text: // Copyright 2018 The Abseil Authors.
  //
  // Licensed under the Apache License, Version 2.0 (the "License");
  // you may not use this file except in compliance with the License.
  // You may obtain a copy of the License at
  //
  //      https://www.apache.org/licenses/LICENSE-2.0
  //
  // Unless required by applicable law or agreed to in writing, software
  // distributed under the License is distributed on an "AS IS" BASIS,
  // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  // See the License for the specific language governing permissions and
  // limitations under the License.
  
  // A btree implementation of the STL set and map interfaces. A btree is smaller
  // and generally also faster than STL set/map (refer to the benchmarks below).
  // The red-black tree implementation of STL set/map has an overhead of 3
  // pointers (left, right and parent) plus the node color information for each
  // stored value. So a set<int32_t> consumes 40 bytes for each value stored in
  // 64-bit mode. This btree implementation stores multiple values on fixed
  // size nodes (usually 256 bytes) and doesn't store child pointers for leaf
  // nodes. The result is that a btree_set<int32_t> may use much less memory per
  // stored value. For the random insertion benchmark in btree_bench.cc, a
  // btree_set<int32_t> with node-size of 256 uses 5.1 bytes per stored value.
  //
  // The packing of multiple values on to each node of a btree has another effect
  // besides better space utilization: better cache locality due to fewer cache
  // lines being accessed. Better cache locality translates into faster
  // operations.
  //
  // CAVEATS
  //
  // Insertions and deletions on a btree can cause splitting, merging or
  // rebalancing of btree nodes. And even without these operations, insertions
  // and deletions on a btree will move values around within a node. In both
  // cases, the result is that insertions and deletions can invalidate iterators
  // pointing to values other than the one being inserted/deleted. Therefore, this
  // container does not provide pointer stability. This is notably different from
  // STL set/map which takes care to not invalidate iterators on insert/erase
  // except, of course, for iterators pointing to the value being erased.  A
  // partial workaround when erasing is available: erase() returns an iterator
  // pointing to the item just after the one that was erased (or end() if none
  // exists).
  
  #ifndef ABSL_CONTAINER_INTERNAL_BTREE_H_
  #define ABSL_CONTAINER_INTERNAL_BTREE_H_
  
  #include <algorithm>
  #include <cassert>
  #include <cstddef>
  #include <cstdint>
  #include <cstring>
  #include <functional>
  #include <iterator>
  #include <limits>
  #include <new>
  #include <string>
  #include <type_traits>
  #include <utility>
  
  #include "absl/base/macros.h"
  #include "absl/container/internal/common.h"
  #include "absl/container/internal/compressed_tuple.h"
  #include "absl/container/internal/container_memory.h"
  #include "absl/container/internal/layout.h"
  #include "absl/memory/memory.h"
  #include "absl/meta/type_traits.h"
  #include "absl/strings/cord.h"
  #include "absl/strings/string_view.h"
  #include "absl/types/compare.h"
  #include "absl/utility/utility.h"
  
  namespace absl {
  ABSL_NAMESPACE_BEGIN
  namespace container_internal {
  
  // A helper class that indicates if the Compare parameter is a key-compare-to
  // comparator.
  #if !defined(__cpp_lib_is_invocable)
  template <typename Compare, typename T>
  using btree_is_key_compare_to =
      std::is_convertible<absl::result_of_t<Compare(const T &, const T &)>,
                          absl::weak_ordering>;
  #else
  template <typename Compare, typename T>
  using btree_is_key_compare_to =
      std::is_convertible<absl::invoke_result_t<Compare, const T &, const T &>,
                          absl::weak_ordering>;
  #endif
  
  struct StringBtreeDefaultLess {
    using is_transparent = void;
  
    StringBtreeDefaultLess() = default;
  
    // Compatibility constructor.
    StringBtreeDefaultLess(std::less<std::string>) {}  // NOLINT
    StringBtreeDefaultLess(std::less<string_view>) {}  // NOLINT
  
    absl::weak_ordering operator()(absl::string_view lhs,
                                   absl::string_view rhs) const {
      return compare_internal::compare_result_as_ordering(lhs.compare(rhs));
    }
    StringBtreeDefaultLess(std::less<absl::Cord>) {}  // NOLINT
    absl::weak_ordering operator()(const absl::Cord &lhs,
                                   const absl::Cord &rhs) const {
      return compare_internal::compare_result_as_ordering(lhs.Compare(rhs));
    }
    absl::weak_ordering operator()(const absl::Cord &lhs,
                                   absl::string_view rhs) const {
      return compare_internal::compare_result_as_ordering(lhs.Compare(rhs));
    }
    absl::weak_ordering operator()(absl::string_view lhs,
                                   const absl::Cord &rhs) const {
      return compare_internal::compare_result_as_ordering(-rhs.Compare(lhs));
    }
  };
  
  struct StringBtreeDefaultGreater {
    using is_transparent = void;
  
    StringBtreeDefaultGreater() = default;
  
    StringBtreeDefaultGreater(std::greater<std::string>) {}  // NOLINT
    StringBtreeDefaultGreater(std::greater<string_view>) {}  // NOLINT
  
    absl::weak_ordering operator()(absl::string_view lhs,
                                   absl::string_view rhs) const {
      return compare_internal::compare_result_as_ordering(rhs.compare(lhs));
    }
    StringBtreeDefaultGreater(std::greater<absl::Cord>) {}  // NOLINT
    absl::weak_ordering operator()(const absl::Cord &lhs,
                                   const absl::Cord &rhs) const {
      return compare_internal::compare_result_as_ordering(rhs.Compare(lhs));
    }
    absl::weak_ordering operator()(const absl::Cord &lhs,
                                   absl::string_view rhs) const {
      return compare_internal::compare_result_as_ordering(-lhs.Compare(rhs));
    }
    absl::weak_ordering operator()(absl::string_view lhs,
                                   const absl::Cord &rhs) const {
      return compare_internal::compare_result_as_ordering(rhs.Compare(lhs));
    }
  };
  
  // A helper class to convert a boolean comparison into a three-way "compare-to"
  // comparison that returns a negative value to indicate less-than, zero to
  // indicate equality and a positive value to indicate greater-than. This helper
  // class is specialized for less<std::string>, greater<std::string>,
  // less<string_view>, greater<string_view>, less<absl::Cord>, and
  // greater<absl::Cord>.
  //
  // key_compare_to_adapter is provided so that btree users
  // automatically get the more efficient compare-to code when using common
  // google string types with common comparison functors.
  // These string-like specializations also turn on heterogeneous lookup by
  // default.
  template <typename Compare>
  struct key_compare_to_adapter {
    using type = Compare;
  };
  
  template <>
  struct key_compare_to_adapter<std::less<std::string>> {
    using type = StringBtreeDefaultLess;
  };
  
  template <>
  struct key_compare_to_adapter<std::greater<std::string>> {
    using type = StringBtreeDefaultGreater;
  };
  
  template <>
  struct key_compare_to_adapter<std::less<absl::string_view>> {
    using type = StringBtreeDefaultLess;
  };
  
  template <>
  struct key_compare_to_adapter<std::greater<absl::string_view>> {
    using type = StringBtreeDefaultGreater;
  };
  
  template <>
  struct key_compare_to_adapter<std::less<absl::Cord>> {
    using type = StringBtreeDefaultLess;
  };
  
  template <>
  struct key_compare_to_adapter<std::greater<absl::Cord>> {
    using type = StringBtreeDefaultGreater;
  };
  
  template <typename Key, typename Compare, typename Alloc, int TargetNodeSize,
            bool Multi, typename SlotPolicy>
  struct common_params {
    // If Compare is a common comparator for a string-like type, then we adapt it
    // to use heterogeneous lookup and to be a key-compare-to comparator.
    using key_compare = typename key_compare_to_adapter<Compare>::type;
    // A type which indicates if we have a key-compare-to functor or a plain old
    // key-compare functor.
    using is_key_compare_to = btree_is_key_compare_to<key_compare, Key>;
  
    using allocator_type = Alloc;
    using key_type = Key;
    using size_type = std::make_signed<size_t>::type;
    using difference_type = ptrdiff_t;
  
    // True if this is a multiset or multimap.
    using is_multi_container = std::integral_constant<bool, Multi>;
  
    using slot_policy = SlotPolicy;
    using slot_type = typename slot_policy::slot_type;
    using value_type = typename slot_policy::value_type;
    using init_type = typename slot_policy::mutable_value_type;
    using pointer = value_type *;
    using const_pointer = const value_type *;
    using reference = value_type &;
    using const_reference = const value_type &;
  
    enum {
      kTargetNodeSize = TargetNodeSize,
  
      // Upper bound for the available space for values. This is largest for leaf
      // nodes, which have overhead of at least a pointer + 4 bytes (for storing
      // 3 field_types and an enum).
      kNodeValueSpace =
          TargetNodeSize - /*minimum overhead=*/(sizeof(void *) + 4),
    };
  
    // This is an integral type large enough to hold as many
    // ValueSize-values as will fit a node of TargetNodeSize bytes.
    using node_count_type =
        absl::conditional_t<(kNodeValueSpace / sizeof(value_type) >
                             (std::numeric_limits<uint8_t>::max)()),
                            uint16_t, uint8_t>;  // NOLINT
  
    // The following methods are necessary for passing this struct as PolicyTraits
    // for node_handle and/or are used within btree.
    static value_type &element(slot_type *slot) {
      return slot_policy::element(slot);
    }
    static const value_type &element(const slot_type *slot) {
      return slot_policy::element(slot);
    }
    template <class... Args>
    static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
      slot_policy::construct(alloc, slot, std::forward<Args>(args)...);
    }
    static void construct(Alloc *alloc, slot_type *slot, slot_type *other) {
      slot_policy::construct(alloc, slot, other);
    }
    static void destroy(Alloc *alloc, slot_type *slot) {
      slot_policy::destroy(alloc, slot);
    }
    static void transfer(Alloc *alloc, slot_type *new_slot, slot_type *old_slot) {
      construct(alloc, new_slot, old_slot);
      destroy(alloc, old_slot);
    }
    static void swap(Alloc *alloc, slot_type *a, slot_type *b) {
      slot_policy::swap(alloc, a, b);
    }
    static void move(Alloc *alloc, slot_type *src, slot_type *dest) {
      slot_policy::move(alloc, src, dest);
    }
    static void move(Alloc *alloc, slot_type *first, slot_type *last,
                     slot_type *result) {
      slot_policy::move(alloc, first, last, result);
    }
  };
  
  // A parameters structure for holding the type parameters for a btree_map.
  // Compare and Alloc should be nothrow copy-constructible.
  template <typename Key, typename Data, typename Compare, typename Alloc,
            int TargetNodeSize, bool Multi>
  struct map_params : common_params<Key, Compare, Alloc, TargetNodeSize, Multi,
                                    map_slot_policy<Key, Data>> {
    using super_type = typename map_params::common_params;
    using mapped_type = Data;
    // This type allows us to move keys when it is safe to do so. It is safe
    // for maps in which value_type and mutable_value_type are layout compatible.
    using slot_policy = typename super_type::slot_policy;
    using slot_type = typename super_type::slot_type;
    using value_type = typename super_type::value_type;
    using init_type = typename super_type::init_type;
  
    using key_compare = typename super_type::key_compare;
    // Inherit from key_compare for empty base class optimization.
    struct value_compare : private key_compare {
      value_compare() = default;
      explicit value_compare(const key_compare &cmp) : key_compare(cmp) {}
  
      template <typename T, typename U>
      auto operator()(const T &left, const U &right) const
          -> decltype(std::declval<key_compare>()(left.first, right.first)) {
        return key_compare::operator()(left.first, right.first);
      }
    };
    using is_map_container = std::true_type;
  
    static const Key &key(const value_type &value) { return value.first; }
    static const Key &key(const init_type &init) { return init.first; }
    static const Key &key(const slot_type *s) { return slot_policy::key(s); }
    static mapped_type &value(value_type *value) { return value->second; }
  };
  
  // This type implements the necessary functions from the
  // absl::container_internal::slot_type interface.
  template <typename Key>
  struct set_slot_policy {
    using slot_type = Key;
    using value_type = Key;
    using mutable_value_type = Key;
  
    static value_type &element(slot_type *slot) { return *slot; }
    static const value_type &element(const slot_type *slot) { return *slot; }
  
    template <typename Alloc, class... Args>
    static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
      absl::allocator_traits<Alloc>::construct(*alloc, slot,
                                               std::forward<Args>(args)...);
    }
  
    template <typename Alloc>
    static void construct(Alloc *alloc, slot_type *slot, slot_type *other) {
      absl::allocator_traits<Alloc>::construct(*alloc, slot, std::move(*other));
    }
  
    template <typename Alloc>
    static void destroy(Alloc *alloc, slot_type *slot) {
      absl::allocator_traits<Alloc>::destroy(*alloc, slot);
    }
  
    template <typename Alloc>
    static void swap(Alloc * /*alloc*/, slot_type *a, slot_type *b) {
      using std::swap;
      swap(*a, *b);
    }
  
    template <typename Alloc>
    static void move(Alloc * /*alloc*/, slot_type *src, slot_type *dest) {
      *dest = std::move(*src);
    }
  
    template <typename Alloc>
    static void move(Alloc *alloc, slot_type *first, slot_type *last,
                     slot_type *result) {
      for (slot_type *src = first, *dest = result; src != last; ++src, ++dest)
        move(alloc, src, dest);
    }
  };
  
  // A parameters structure for holding the type parameters for a btree_set.
  // Compare and Alloc should be nothrow copy-constructible.
  template <typename Key, typename Compare, typename Alloc, int TargetNodeSize,
            bool Multi>
  struct set_params : common_params<Key, Compare, Alloc, TargetNodeSize, Multi,
                                    set_slot_policy<Key>> {
    using value_type = Key;
    using slot_type = typename set_params::common_params::slot_type;
    using value_compare = typename set_params::common_params::key_compare;
    using is_map_container = std::false_type;
  
    static const Key &key(const value_type &value) { return value; }
    static const Key &key(const slot_type *slot) { return *slot; }
  };
  
  // An adapter class that converts a lower-bound compare into an upper-bound
  // compare. Note: there is no need to make a version of this adapter specialized
  // for key-compare-to functors because the upper-bound (the first value greater
  // than the input) is never an exact match.
  template <typename Compare>
  struct upper_bound_adapter {
    explicit upper_bound_adapter(const Compare &c) : comp(c) {}
    template <typename K1, typename K2>
    bool operator()(const K1 &a, const K2 &b) const {
      // Returns true when a is not greater than b.
      return !compare_internal::compare_result_as_less_than(comp(b, a));
    }
  
   private:
    Compare comp;
  };
  
  enum class MatchKind : uint8_t { kEq, kNe };
  
  template <typename V, bool IsCompareTo>
  struct SearchResult {
    V value;
    MatchKind match;
  
    static constexpr bool HasMatch() { return true; }
    bool IsEq() const { return match == MatchKind::kEq; }
  };
  
  // When we don't use CompareTo, `match` is not present.
  // This ensures that callers can't use it accidentally when it provides no
  // useful information.
  template <typename V>
  struct SearchResult<V, false> {
    V value;
  
    static constexpr bool HasMatch() { return false; }
    static constexpr bool IsEq() { return false; }
  };
  
  // A node in the btree holding. The same node type is used for both internal
  // and leaf nodes in the btree, though the nodes are allocated in such a way
  // that the children array is only valid in internal nodes.
  template <typename Params>
  class btree_node {
    using is_key_compare_to = typename Params::is_key_compare_to;
    using is_multi_container = typename Params::is_multi_container;
    using field_type = typename Params::node_count_type;
    using allocator_type = typename Params::allocator_type;
    using slot_type = typename Params::slot_type;
  
   public:
    using params_type = Params;
    using key_type = typename Params::key_type;
    using value_type = typename Params::value_type;
    using pointer = typename Params::pointer;
    using const_pointer = typename Params::const_pointer;
    using reference = typename Params::reference;
    using const_reference = typename Params::const_reference;
    using key_compare = typename Params::key_compare;
    using size_type = typename Params::size_type;
    using difference_type = typename Params::difference_type;
  
    // Btree decides whether to use linear node search as follows:
    //   - If the key is arithmetic and the comparator is std::less or
    //     std::greater, choose linear.
    //   - Otherwise, choose binary.
    // TODO(ezb): Might make sense to add condition(s) based on node-size.
    using use_linear_search = std::integral_constant<
        bool, std::is_arithmetic<key_type>::value &&
                  (std::is_same<std::less<key_type>, key_compare>::value ||
                   std::is_same<std::greater<key_type>, key_compare>::value)>;
  
    // This class is organized by gtl::Layout as if it had the following
    // structure:
    //   // A pointer to the node's parent.
    //   btree_node *parent;
    //
    //   // The position of the node in the node's parent.
    //   field_type position;
    //   // The index of the first populated value in `values`.
    //   // TODO(ezb): right now, `start` is always 0. Update insertion/merge
    //   // logic to allow for floating storage within nodes.
    //   field_type start;
    //   // The index after the last populated value in `values`. Currently, this
    //   // is the same as the count of values.
    //   field_type finish;
    //   // The maximum number of values the node can hold. This is an integer in
    //   // [1, kNodeValues] for root leaf nodes, kNodeValues for non-root leaf
    //   // nodes, and kInternalNodeMaxCount (as a sentinel value) for internal
    //   // nodes (even though there are still kNodeValues values in the node).
    //   // TODO(ezb): make max_count use only 4 bits and record log2(capacity)
    //   // to free extra bits for is_root, etc.
    //   field_type max_count;
    //
    //   // The array of values. The capacity is `max_count` for leaf nodes and
    //   // kNodeValues for internal nodes. Only the values in
    //   // [start, finish) have been initialized and are valid.
    //   slot_type values[max_count];
    //
    //   // The array of child pointers. The keys in children[i] are all less
    //   // than key(i). The keys in children[i + 1] are all greater than key(i).
    //   // There are 0 children for leaf nodes and kNodeValues + 1 children for
    //   // internal nodes.
    //   btree_node *children[kNodeValues + 1];
    //
    // This class is only constructed by EmptyNodeType. Normally, pointers to the
    // layout above are allocated, cast to btree_node*, and de-allocated within
    // the btree implementation.
    ~btree_node() = default;
    btree_node(btree_node const &) = delete;
    btree_node &operator=(btree_node const &) = delete;
  
    // Public for EmptyNodeType.
    constexpr static size_type Alignment() {
      static_assert(LeafLayout(1).Alignment() == InternalLayout().Alignment(),
                    "Alignment of all nodes must be equal.");
      return InternalLayout().Alignment();
    }
  
   protected:
    btree_node() = default;
  
   private:
    using layout_type = absl::container_internal::Layout<btree_node *, field_type,
                                                         slot_type, btree_node *>;
    constexpr static size_type SizeWithNValues(size_type n) {
      return layout_type(/*parent*/ 1,
                         /*position, start, finish, max_count*/ 4,
                         /*values*/ n,
                         /*children*/ 0)
          .AllocSize();
    }
    // A lower bound for the overhead of fields other than values in a leaf node.
    constexpr static size_type MinimumOverhead() {
      return SizeWithNValues(1) - sizeof(value_type);
    }
  
    // Compute how many values we can fit onto a leaf node taking into account
    // padding.
    constexpr static size_type NodeTargetValues(const int begin, const int end) {
      return begin == end ? begin
                          : SizeWithNValues((begin + end) / 2 + 1) >
                                    params_type::kTargetNodeSize
                                ? NodeTargetValues(begin, (begin + end) / 2)
                                : NodeTargetValues((begin + end) / 2 + 1, end);
    }
  
    enum {
      kTargetNodeSize = params_type::kTargetNodeSize,
      kNodeTargetValues = NodeTargetValues(0, params_type::kTargetNodeSize),
  
      // We need a minimum of 3 values per internal node in order to perform
      // splitting (1 value for the two nodes involved in the split and 1 value
      // propagated to the parent as the delimiter for the split).
      kNodeValues = kNodeTargetValues >= 3 ? kNodeTargetValues : 3,
  
      // The node is internal (i.e. is not a leaf node) if and only if `max_count`
      // has this value.
      kInternalNodeMaxCount = 0,
    };
  
    // Leaves can have less than kNodeValues values.
    constexpr static layout_type LeafLayout(const int max_values = kNodeValues) {
      return layout_type(/*parent*/ 1,
                         /*position, start, finish, max_count*/ 4,
                         /*values*/ max_values,
                         /*children*/ 0);
    }
    constexpr static layout_type InternalLayout() {
      return layout_type(/*parent*/ 1,
                         /*position, start, finish, max_count*/ 4,
                         /*values*/ kNodeValues,
                         /*children*/ kNodeValues + 1);
    }
    constexpr static size_type LeafSize(const int max_values = kNodeValues) {
      return LeafLayout(max_values).AllocSize();
    }
    constexpr static size_type InternalSize() {
      return InternalLayout().AllocSize();
    }
  
    // N is the index of the type in the Layout definition.
    // ElementType<N> is the Nth type in the Layout definition.
    template <size_type N>
    inline typename layout_type::template ElementType<N> *GetField() {
      // We assert that we don't read from values that aren't there.
      assert(N < 3 || !leaf());
      return InternalLayout().template Pointer<N>(reinterpret_cast<char *>(this));
    }
    template <size_type N>
    inline const typename layout_type::template ElementType<N> *GetField() const {
      assert(N < 3 || !leaf());
      return InternalLayout().template Pointer<N>(
          reinterpret_cast<const char *>(this));
    }
    void set_parent(btree_node *p) { *GetField<0>() = p; }
    field_type &mutable_finish() { return GetField<1>()[2]; }
    slot_type *slot(int i) { return &GetField<2>()[i]; }
    slot_type *start_slot() { return slot(start()); }
    slot_type *finish_slot() { return slot(finish()); }
    const slot_type *slot(int i) const { return &GetField<2>()[i]; }
    void set_position(field_type v) { GetField<1>()[0] = v; }
    void set_start(field_type v) { GetField<1>()[1] = v; }
    void set_finish(field_type v) { GetField<1>()[2] = v; }
    // This method is only called by the node init methods.
    void set_max_count(field_type v) { GetField<1>()[3] = v; }
  
   public:
    // Whether this is a leaf node or not. This value doesn't change after the
    // node is created.
    bool leaf() const { return GetField<1>()[3] != kInternalNodeMaxCount; }
  
    // Getter for the position of this node in its parent.
    field_type position() const { return GetField<1>()[0]; }
  
    // Getter for the offset of the first value in the `values` array.
    field_type start() const {
      // TODO(ezb): when floating storage is implemented, return GetField<1>()[1];
      assert(GetField<1>()[1] == 0);
      return 0;
    }
  
    // Getter for the offset after the last value in the `values` array.
    field_type finish() const { return GetField<1>()[2]; }
  
    // Getters for the number of values stored in this node.
    field_type count() const {
      assert(finish() >= start());
      return finish() - start();
    }
    field_type max_count() const {
      // Internal nodes have max_count==kInternalNodeMaxCount.
      // Leaf nodes have max_count in [1, kNodeValues].
      const field_type max_count = GetField<1>()[3];
      return max_count == field_type{kInternalNodeMaxCount}
                 ? field_type{kNodeValues}
                 : max_count;
    }
  
    // Getter for the parent of this node.
    btree_node *parent() const { return *GetField<0>(); }
    // Getter for whether the node is the root of the tree. The parent of the
    // root of the tree is the leftmost node in the tree which is guaranteed to
    // be a leaf.
    bool is_root() const { return parent()->leaf(); }
    void make_root() {
      assert(parent()->is_root());
      set_parent(parent()->parent());
    }
  
    // Getters for the key/value at position i in the node.
    const key_type &key(int i) const { return params_type::key(slot(i)); }
    reference value(int i) { return params_type::element(slot(i)); }
    const_reference value(int i) const { return params_type::element(slot(i)); }
  
    // Getters/setter for the child at position i in the node.
    btree_node *child(int i) const { return GetField<3>()[i]; }
    btree_node *start_child() const { return child(start()); }
    btree_node *&mutable_child(int i) { return GetField<3>()[i]; }
    void clear_child(int i) {
      absl::container_internal::SanitizerPoisonObject(&mutable_child(i));
    }
    void set_child(int i, btree_node *c) {
      absl::container_internal::SanitizerUnpoisonObject(&mutable_child(i));
      mutable_child(i) = c;
      c->set_position(i);
    }
    void init_child(int i, btree_node *c) {
      set_child(i, c);
      c->set_parent(this);
    }
  
    // Returns the position of the first value whose key is not less than k.
    template <typename K>
    SearchResult<int, is_key_compare_to::value> lower_bound(
        const K &k, const key_compare &comp) const {
      return use_linear_search::value ? linear_search(k, comp)
                                      : binary_search(k, comp);
    }
    // Returns the position of the first value whose key is greater than k.
    template <typename K>
    int upper_bound(const K &k, const key_compare &comp) const {
      auto upper_compare = upper_bound_adapter<key_compare>(comp);
      return use_linear_search::value ? linear_search(k, upper_compare).value
                                      : binary_search(k, upper_compare).value;
    }
  
    template <typename K, typename Compare>
    SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
    linear_search(const K &k, const Compare &comp) const {
      return linear_search_impl(k, start(), finish(), comp,
                                btree_is_key_compare_to<Compare, key_type>());
    }
  
    template <typename K, typename Compare>
    SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
    binary_search(const K &k, const Compare &comp) const {
      return binary_search_impl(k, start(), finish(), comp,
                                btree_is_key_compare_to<Compare, key_type>());
    }
  
    // Returns the position of the first value whose key is not less than k using
    // linear search performed using plain compare.
    template <typename K, typename Compare>
    SearchResult<int, false> linear_search_impl(
        const K &k, int s, const int e, const Compare &comp,
        std::false_type /* IsCompareTo */) const {
      while (s < e) {
        if (!comp(key(s), k)) {
          break;
        }
        ++s;
      }
      return {s};
    }
  
    // Returns the position of the first value whose key is not less than k using
    // linear search performed using compare-to.
    template <typename K, typename Compare>
    SearchResult<int, true> linear_search_impl(
        const K &k, int s, const int e, const Compare &comp,
        std::true_type /* IsCompareTo */) const {
      while (s < e) {
        const absl::weak_ordering c = comp(key(s), k);
        if (c == 0) {
          return {s, MatchKind::kEq};
        } else if (c > 0) {
          break;
        }
        ++s;
      }
      return {s, MatchKind::kNe};
    }
  
    // Returns the position of the first value whose key is not less than k using
    // binary search performed using plain compare.
    template <typename K, typename Compare>
    SearchResult<int, false> binary_search_impl(
        const K &k, int s, int e, const Compare &comp,
        std::false_type /* IsCompareTo */) const {
      while (s != e) {
        const int mid = (s + e) >> 1;
        if (comp(key(mid), k)) {
          s = mid + 1;
        } else {
          e = mid;
        }
      }
      return {s};
    }
  
    // Returns the position of the first value whose key is not less than k using
    // binary search performed using compare-to.
    template <typename K, typename CompareTo>
    SearchResult<int, true> binary_search_impl(
        const K &k, int s, int e, const CompareTo &comp,
        std::true_type /* IsCompareTo */) const {
      if (is_multi_container::value) {
        MatchKind exact_match = MatchKind::kNe;
        while (s != e) {
          const int mid = (s + e) >> 1;
          const absl::weak_ordering c = comp(key(mid), k);
          if (c < 0) {
            s = mid + 1;
          } else {
            e = mid;
            if (c == 0) {
              // Need to return the first value whose key is not less than k,
              // which requires continuing the binary search if this is a
              // multi-container.
              exact_match = MatchKind::kEq;
            }
          }
        }
        return {s, exact_match};
      } else {  // Not a multi-container.
        while (s != e) {
          const int mid = (s + e) >> 1;
          const absl::weak_ordering c = comp(key(mid), k);
          if (c < 0) {
            s = mid + 1;
          } else if (c > 0) {
            e = mid;
          } else {
            return {mid, MatchKind::kEq};
          }
        }
        return {s, MatchKind::kNe};
      }
    }
  
    // Emplaces a value at position i, shifting all existing values and
    // children at positions >= i to the right by 1.
    template <typename... Args>
    void emplace_value(size_type i, allocator_type *alloc, Args &&... args);
  
    // Removes the value at position i, shifting all existing values and children
    // at positions > i to the left by 1.
    void remove_value(int i, allocator_type *alloc);
  
    // Removes the values at positions [i, i + to_erase), shifting all values
    // after that range to the left by to_erase. Does not change children at all.
    void remove_values_ignore_children(int i, int to_erase,
                                       allocator_type *alloc);
  
    // Rebalances a node with its right sibling.
    void rebalance_right_to_left(int to_move, btree_node *right,
                                 allocator_type *alloc);
    void rebalance_left_to_right(int to_move, btree_node *right,
                                 allocator_type *alloc);
  
    // Splits a node, moving a portion of the node's values to its right sibling.
    void split(int insert_position, btree_node *dest, allocator_type *alloc);
  
    // Merges a node with its right sibling, moving all of the values and the
    // delimiting key in the parent node onto itself.
    void merge(btree_node *src, allocator_type *alloc);
  
    // Node allocation/deletion routines.
    void init_leaf(btree_node *parent, int max_count) {
      set_parent(parent);
      set_position(0);
      set_start(0);
      set_finish(0);
      set_max_count(max_count);
      absl::container_internal::SanitizerPoisonMemoryRegion(
          start_slot(), max_count * sizeof(slot_type));
    }
    void init_internal(btree_node *parent) {
      init_leaf(parent, kNodeValues);
      // Set `max_count` to a sentinel value to indicate that this node is
      // internal.
      set_max_count(kInternalNodeMaxCount);
      absl::container_internal::SanitizerPoisonMemoryRegion(
          &mutable_child(start()), (kNodeValues + 1) * sizeof(btree_node *));
    }
    void destroy(allocator_type *alloc) {
      for (int i = start(); i < finish(); ++i) {
        value_destroy(i, alloc);
      }
    }
  
   public:
    // Exposed only for tests.
    static bool testonly_uses_linear_node_search() {
      return use_linear_search::value;
    }
  
   private:
    template <typename... Args>
    void value_init(const size_type i, allocator_type *alloc, Args &&... args) {
      absl::container_internal::SanitizerUnpoisonObject(slot(i));
      params_type::construct(alloc, slot(i), std::forward<Args>(args)...);
    }
    void value_destroy(const size_type i, allocator_type *alloc) {
      params_type::destroy(alloc, slot(i));
      absl::container_internal::SanitizerPoisonObject(slot(i));
    }
  
    // Transfers value from slot `src_i` in `src_node` to slot `dest_i` in `this`.
    void transfer(const size_type dest_i, const size_type src_i,
                  btree_node *src_node, allocator_type *alloc) {
      absl::container_internal::SanitizerUnpoisonObject(slot(dest_i));
      params_type::transfer(alloc, slot(dest_i), src_node->slot(src_i));
      absl::container_internal::SanitizerPoisonObject(src_node->slot(src_i));
    }
  
    // Transfers `n` values starting at value `src_i` in `src_node` into the
    // values starting at value `dest_i` in `this`.
    void transfer_n(const size_type n, const size_type dest_i,
                    const size_type src_i, btree_node *src_node,
                    allocator_type *alloc) {
      absl::container_internal::SanitizerUnpoisonMemoryRegion(
          slot(dest_i), n * sizeof(slot_type));
      for (slot_type *src = src_node->slot(src_i), *end = src + n,
                     *dest = slot(dest_i);
           src != end; ++src, ++dest) {
        params_type::transfer(alloc, dest, src);
      }
      // We take care to avoid poisoning transferred-to nodes in case of overlap.
      const size_type overlap =
          this == src_node ? (std::max)(src_i, dest_i + n) - src_i : 0;
      assert(n >= overlap);
      absl::container_internal::SanitizerPoisonMemoryRegion(
          src_node->slot(src_i + overlap), (n - overlap) * sizeof(slot_type));
    }
  
    // Same as above, except that we start at the end and work our way to the
    // beginning.
    void transfer_n_backward(const size_type n, const size_type dest_i,
                             const size_type src_i, btree_node *src_node,
                             allocator_type *alloc) {
      absl::container_internal::SanitizerUnpoisonMemoryRegion(
          slot(dest_i), n * sizeof(slot_type));
      for (slot_type *src = src_node->slot(src_i + n - 1), *end = src - n,
                     *dest = slot(dest_i + n - 1);
           src != end; --src, --dest) {
        params_type::transfer(alloc, dest, src);
      }
      // We take care to avoid poisoning transferred-to nodes in case of overlap.
      assert(this != src_node || dest_i >= src_i);
      const size_type num_to_poison =
          this == src_node ? (std::min)(n, dest_i - src_i) : n;
      absl::container_internal::SanitizerPoisonMemoryRegion(
          src_node->slot(src_i), num_to_poison * sizeof(slot_type));
    }
  
    template <typename P>
    friend class btree;
    template <typename N, typename R, typename P>
    friend struct btree_iterator;
    friend class BtreeNodePeer;
  };
  
  template <typename Node, typename Reference, typename Pointer>
  struct btree_iterator {
   private:
    using key_type = typename Node::key_type;
    using size_type = typename Node::size_type;
    using params_type = typename Node::params_type;
  
    using node_type = Node;
    using normal_node = typename std::remove_const<Node>::type;
    using const_node = const Node;
    using normal_pointer = typename params_type::pointer;
    using normal_reference = typename params_type::reference;
    using const_pointer = typename params_type::const_pointer;
    using const_reference = typename params_type::const_reference;
    using slot_type = typename params_type::slot_type;
  
    using iterator =
        btree_iterator<normal_node, normal_reference, normal_pointer>;
    using const_iterator =
        btree_iterator<const_node, const_reference, const_pointer>;
  
   public:
    // These aliases are public for std::iterator_traits.
    using difference_type = typename Node::difference_type;
    using value_type = typename params_type::value_type;
    using pointer = Pointer;
    using reference = Reference;
    using iterator_category = std::bidirectional_iterator_tag;
  
    btree_iterator() : node(nullptr), position(-1) {}
    explicit btree_iterator(Node *n) : node(n), position(n->start()) {}
    btree_iterator(Node *n, int p) : node(n), position(p) {}
  
    // NOTE: this SFINAE allows for implicit conversions from iterator to
    // const_iterator, but it specifically avoids defining copy constructors so
    // that btree_iterator can be trivially copyable. This is for performance and
    // binary size reasons.
    template <typename N, typename R, typename P,
              absl::enable_if_t<
                  std::is_same<btree_iterator<N, R, P>, iterator>::value &&
                      std::is_same<btree_iterator, const_iterator>::value,
                  int> = 0>
    btree_iterator(const btree_iterator<N, R, P> &other)  // NOLINT
        : node(other.node), position(other.position) {}
  
   private:
    // This SFINAE allows explicit conversions from const_iterator to
    // iterator, but also avoids defining a copy constructor.
    // NOTE: the const_cast is safe because this constructor is only called by
    // non-const methods and the container owns the nodes.
    template <typename N, typename R, typename P,
              absl::enable_if_t<
                  std::is_same<btree_iterator<N, R, P>, const_iterator>::value &&
                      std::is_same<btree_iterator, iterator>::value,
                  int> = 0>
    explicit btree_iterator(const btree_iterator<N, R, P> &other)
        : node(const_cast<node_type *>(other.node)), position(other.position) {}
  
    // Increment/decrement the iterator.
    void increment() {
      if (node->leaf() && ++position < node->finish()) {
        return;
      }
      increment_slow();
    }
    void increment_slow();
  
    void decrement() {
      if (node->leaf() && --position >= node->start()) {
        return;
      }
      decrement_slow();
    }
    void decrement_slow();
  
   public:
    bool operator==(const iterator &other) const {
      return node == other.node && position == other.position;
    }
    bool operator==(const const_iterator &other) const {
      return node == other.node && position == other.position;
    }
    bool operator!=(const iterator &other) const {
      return node != other.node || position != other.position;
    }
    bool operator!=(const const_iterator &other) const {
      return node != other.node || position != other.position;
    }
  
    // Accessors for the key/value the iterator is pointing at.
    reference operator*() const {
      ABSL_HARDENING_ASSERT(node != nullptr);
      ABSL_HARDENING_ASSERT(node->start() <= position);
      ABSL_HARDENING_ASSERT(node->finish() > position);
      return node->value(position);
    }
    pointer operator->() const { return &operator*(); }
  
    btree_iterator &operator++() {
      increment();
      return *this;
    }
    btree_iterator &operator--() {
      decrement();
      return *this;
    }
    btree_iterator operator++(int) {
      btree_iterator tmp = *this;
      ++*this;
      return tmp;
    }
    btree_iterator operator--(int) {
      btree_iterator tmp = *this;
      --*this;
      return tmp;
    }
  
   private:
    template <typename Params>
    friend class btree;
    template <typename Tree>
    friend class btree_container;
    template <typename Tree>
    friend class btree_set_container;
    template <typename Tree>
    friend class btree_map_container;
    template <typename Tree>
    friend class btree_multiset_container;
    template <typename N, typename R, typename P>
    friend struct btree_iterator;
    template <typename TreeType, typename CheckerType>
    friend class base_checker;
  
    const key_type &key() const { return node->key(position); }
    slot_type *slot() { return node->slot(position); }
  
    // The node in the tree the iterator is pointing at.
    Node *node;
    // The position within the node of the tree the iterator is pointing at.
    // NOTE: this is an int rather than a field_type because iterators can point
    // to invalid positions (such as -1) in certain circumstances.
    int position;
  };
  
  template <typename Params>
  class btree {
    using node_type = btree_node<Params>;
    using is_key_compare_to = typename Params::is_key_compare_to;
  
    // We use a static empty node for the root/leftmost/rightmost of empty btrees
    // in order to avoid branching in begin()/end().
    struct alignas(node_type::Alignment()) EmptyNodeType : node_type {
      using field_type = typename node_type::field_type;
      node_type *parent;
      field_type position = 0;
      field_type start = 0;
      field_type finish = 0;
      // max_count must be != kInternalNodeMaxCount (so that this node is regarded
      // as a leaf node). max_count() is never called when the tree is empty.
      field_type max_count = node_type::kInternalNodeMaxCount + 1;
  
  #ifdef _MSC_VER
      // MSVC has constexpr code generations bugs here.
      EmptyNodeType() : parent(this) {}
  #else
      constexpr EmptyNodeType(node_type *p) : parent(p) {}
  #endif
    };
  
    static node_type *EmptyNode() {
  #ifdef _MSC_VER
      static EmptyNodeType *empty_node = new EmptyNodeType;
      // This assert fails on some other construction methods.
      assert(empty_node->parent == empty_node);
      return empty_node;
  #else
      static constexpr EmptyNodeType empty_node(
          const_cast<EmptyNodeType *>(&empty_node));
      return const_cast<EmptyNodeType *>(&empty_node);
  #endif
    }
  
    enum {
      kNodeValues = node_type::kNodeValues,
      kMinNodeValues = kNodeValues / 2,
    };
  
    struct node_stats {
      using size_type = typename Params::size_type;
  
      node_stats(size_type l, size_type i) : leaf_nodes(l), internal_nodes(i) {}
  
      node_stats &operator+=(const node_stats &other) {
        leaf_nodes += other.leaf_nodes;
        internal_nodes += other.internal_nodes;
        return *this;
      }
  
      size_type leaf_nodes;
      size_type internal_nodes;
    };
  
   public:
    using key_type = typename Params::key_type;
    using value_type = typename Params::value_type;
    using size_type = typename Params::size_type;
    using difference_type = typename Params::difference_type;
    using key_compare = typename Params::key_compare;
    using value_compare = typename Params::value_compare;
    using allocator_type = typename Params::allocator_type;
    using reference = typename Params::reference;
    using const_reference = typename Params::const_reference;
    using pointer = typename Params::pointer;
    using const_pointer = typename Params::const_pointer;
    using iterator = btree_iterator<node_type, reference, pointer>;
    using const_iterator = typename iterator::const_iterator;
    using reverse_iterator = std::reverse_iterator<iterator>;
    using const_reverse_iterator = std::reverse_iterator<const_iterator>;
    using node_handle_type = node_handle<Params, Params, allocator_type>;
  
    // Internal types made public for use by btree_container types.
    using params_type = Params;
    using slot_type = typename Params::slot_type;
  
   private:
    // For use in copy_or_move_values_in_order.
    const value_type &maybe_move_from_iterator(const_iterator it) { return *it; }
    value_type &&maybe_move_from_iterator(iterator it) { return std::move(*it); }
  
    // Copies or moves (depending on the template parameter) the values in
    // other into this btree in their order in other. This btree must be empty
    // before this method is called. This method is used in copy construction,
    // copy assignment, and move assignment.
    template <typename Btree>
    void copy_or_move_values_in_order(Btree *other);
  
    // Validates that various assumptions/requirements are true at compile time.
    constexpr static bool static_assert_validation();
  
   public:
    btree(const key_compare &comp, const allocator_type &alloc);
  
    btree(const btree &other);
    btree(btree &&other) noexcept
        : root_(std::move(other.root_)),
          rightmost_(absl::exchange(other.rightmost_, EmptyNode())),
          size_(absl::exchange(other.size_, 0)) {
      other.mutable_root() = EmptyNode();
    }
  
    ~btree() {
      // Put static_asserts in destructor to avoid triggering them before the type
      // is complete.
      static_assert(static_assert_validation(), "This call must be elided.");
      clear();
    }
  
    // Assign the contents of other to *this.
    btree &operator=(const btree &other);
    btree &operator=(btree &&other) noexcept;
  
    iterator begin() { return iterator(leftmost()); }
    const_iterator begin() const { return const_iterator(leftmost()); }
    iterator end() { return iterator(rightmost_, rightmost_->finish()); }
    const_iterator end() const {
      return const_iterator(rightmost_, rightmost_->finish());
    }
    reverse_iterator rbegin() { return reverse_iterator(end()); }
    const_reverse_iterator rbegin() const {
      return const_reverse_iterator(end());
    }
    reverse_iterator rend() { return reverse_iterator(begin()); }
    const_reverse_iterator rend() const {
      return const_reverse_iterator(begin());
    }
  
    // Finds the first element whose key is not less than key.
    template <typename K>
    iterator lower_bound(const K &key) {
      return internal_end(internal_lower_bound(key));
    }
    template <typename K>
    const_iterator lower_bound(const K &key) const {
      return internal_end(internal_lower_bound(key));
    }
  
    // Finds the first element whose key is greater than key.
    template <typename K>
    iterator upper_bound(const K &key) {
      return internal_end(internal_upper_bound(key));
    }
    template <typename K>
    const_iterator upper_bound(const K &key) const {
      return internal_end(internal_upper_bound(key));
    }
  
    // Finds the range of values which compare equal to key. The first member of
    // the returned pair is equal to lower_bound(key). The second member pair of
    // the pair is equal to upper_bound(key).
    template <typename K>
    std::pair<iterator, iterator> equal_range(const K &key) {
      return {lower_bound(key), upper_bound(key)};
    }
    template <typename K>
    std::pair<const_iterator, const_iterator> equal_range(const K &key) const {
      return {lower_bound(key), upper_bound(key)};
    }
  
    // Inserts a value into the btree only if it does not already exist. The
    // boolean return value indicates whether insertion succeeded or failed.
    // Requirement: if `key` already exists in the btree, does not consume `args`.
    // Requirement: `key` is never referenced after consuming `args`.
    template <typename... Args>
    std::pair<iterator, bool> insert_unique(const key_type &key, Args &&... args);
  
    // Inserts with hint. Checks to see if the value should be placed immediately
    // before `position` in the tree. If so, then the insertion will take
    // amortized constant time. If not, the insertion will take amortized
    // logarithmic time as if a call to insert_unique() were made.
    // Requirement: if `key` already exists in the btree, does not consume `args`.
    // Requirement: `key` is never referenced after consuming `args`.
    template <typename... Args>
    std::pair<iterator, bool> insert_hint_unique(iterator position,
                                                 const key_type &key,
                                                 Args &&... args);
  
    // Insert a range of values into the btree.
    template <typename InputIterator>
    void insert_iterator_unique(InputIterator b, InputIterator e);
  
    // Inserts a value into the btree.
    template <typename ValueType>
    iterator insert_multi(const key_type &key, ValueType &&v);
  
    // Inserts a value into the btree.
    template <typename ValueType>
    iterator insert_multi(ValueType &&v) {
      return insert_multi(params_type::key(v), std::forward<ValueType>(v));
    }
  
    // Insert with hint. Check to see if the value should be placed immediately
    // before position in the tree. If it does, then the insertion will take
    // amortized constant time. If not, the insertion will take amortized
    // logarithmic time as if a call to insert_multi(v) were made.
    template <typename ValueType>
    iterator insert_hint_multi(iterator position, ValueType &&v);
  
    // Insert a range of values into the btree.
    template <typename InputIterator>
    void insert_iterator_multi(InputIterator b, InputIterator e);
  
    // Erase the specified iterator from the btree. The iterator must be valid
    // (i.e. not equal to end()).  Return an iterator pointing to the node after
    // the one that was erased (or end() if none exists).
    // Requirement: does not read the value at `*iter`.
    iterator erase(iterator iter);
  
    // Erases range. Returns the number of keys erased and an iterator pointing
    // to the element after the last erased element.
    std::pair<size_type, iterator> erase_range(iterator begin, iterator end);
  
    // Erases the specified key from the btree. Returns 1 if an element was
    // erased and 0 otherwise.
    template <typename K>
    size_type erase_unique(const K &key);
  
    // Erases all of the entries matching the specified key from the
    // btree. Returns the number of elements erased.
    template <typename K>
    size_type erase_multi(const K &key);
  
    // Finds the iterator corresponding to a key or returns end() if the key is
    // not present.
    template <typename K>
    iterator find(const K &key) {
      return internal_end(internal_find(key));
    }
    template <typename K>
    const_iterator find(const K &key) const {
      return internal_end(internal_find(key));
    }
  
    // Returns a count of the number of times the key appears in the btree.
    template <typename K>
    size_type count_unique(const K &key) const {
      const iterator begin = internal_find(key);
      if (begin.node == nullptr) {
        // The key doesn't exist in the tree.
        return 0;
      }
      return 1;
    }
    // Returns a count of the number of times the key appears in the btree.
    template <typename K>
    size_type count_multi(const K &key) const {
      const auto range = equal_range(key);
      return std::distance(range.first, range.second);
    }
  
    // Clear the btree, deleting all of the values it contains.
    void clear();
  
    // Swaps the contents of `this` and `other`.
    void swap(btree &other);
  
    const key_compare &key_comp() const noexcept {
      return root_.template get<0>();
    }
    template <typename K1, typename K2>
    bool compare_keys(const K1 &a, const K2 &b) const {
      return compare_internal::compare_result_as_less_than(key_comp()(a, b));
    }
  
    value_compare value_comp() const { return value_compare(key_comp()); }
  
    // Verifies the structure of the btree.
    void verify() const;
  
    // Size routines.
    size_type size() const { return size_; }
    size_type max_size() const { return (std::numeric_limits<size_type>::max)(); }
    bool empty() const { return size_ == 0; }
  
    // The height of the btree. An empty tree will have height 0.
    size_type height() const {
      size_type h = 0;
      if (!empty()) {
        // Count the length of the chain from the leftmost node up to the
        // root. We actually count from the root back around to the level below
        // the root, but the calculation is the same because of the circularity
        // of that traversal.
        const node_type *n = root();
        do {
          ++h;
          n = n->parent();
        } while (n != root());
      }
      return h;
    }
  
    // The number of internal, leaf and total nodes used by the btree.
    size_type leaf_nodes() const { return internal_stats(root()).leaf_nodes; }
    size_type internal_nodes() const {
      return internal_stats(root()).internal_nodes;
    }
    size_type nodes() const {
      node_stats stats = internal_stats(root());
      return stats.leaf_nodes + stats.internal_nodes;
    }
  
    // The total number of bytes used by the btree.
    size_type bytes_used() const {
      node_stats stats = internal_stats(root());
      if (stats.leaf_nodes == 1 && stats.internal_nodes == 0) {
        return sizeof(*this) + node_type::LeafSize(root()->max_count());
      } else {
        return sizeof(*this) + stats.leaf_nodes * node_type::LeafSize() +
               stats.internal_nodes * node_type::InternalSize();
      }
    }
  
    // The average number of bytes used per value stored in the btree.
    static double average_bytes_per_value() {
      // Returns the number of bytes per value on a leaf node that is 75%
      // full. Experimentally, this matches up nicely with the computed number of
      // bytes per value in trees that had their values inserted in random order.
      return node_type::LeafSize() / (kNodeValues * 0.75);
    }
  
    // The fullness of the btree. Computed as the number of elements in the btree
    // divided by the maximum number of elements a tree with the current number
    // of nodes could hold. A value of 1 indicates perfect space
    // utilization. Smaller values indicate space wastage.
    // Returns 0 for empty trees.
    double fullness() const {
      if (empty()) return 0.0;
      return static_cast<double>(size()) / (nodes() * kNodeValues);
    }
    // The overhead of the btree structure in bytes per node. Computed as the
    // total number of bytes used by the btree minus the number of bytes used for
    // storing elements divided by the number of elements.
    // Returns 0 for empty trees.
    double overhead() const {
      if (empty()) return 0.0;
      return (bytes_used() - size() * sizeof(value_type)) /
             static_cast<double>(size());
    }
  
    // The allocator used by the btree.
    allocator_type get_allocator() const { return allocator(); }
  
   private:
    // Internal accessor routines.
    node_type *root() { return root_.template get<2>(); }
    const node_type *root() const { return root_.template get<2>(); }
    node_type *&mutable_root() noexcept { return root_.template get<2>(); }
    key_compare *mutable_key_comp() noexcept { return &root_.template get<0>(); }
  
    // The leftmost node is stored as the parent of the root node.
    node_type *leftmost() { return root()->parent(); }
    const node_type *leftmost() const { return root()->parent(); }
  
    // Allocator routines.
    allocator_type *mutable_allocator() noexcept {
      return &root_.template get<1>();
    }
    const allocator_type &allocator() const noexcept {
      return root_.template get<1>();
    }
  
    // Allocates a correctly aligned node of at least size bytes using the
    // allocator.
    node_type *allocate(const size_type size) {
      return reinterpret_cast<node_type *>(
          absl::container_internal::Allocate<node_type::Alignment()>(
              mutable_allocator(), size));
    }
  
    // Node creation/deletion routines.
    node_type *new_internal_node(node_type *parent) {
      node_type *n = allocate(node_type::InternalSize());
      n->init_internal(parent);
      return n;
    }
    node_type *new_leaf_node(node_type *parent) {
      node_type *n = allocate(node_type::LeafSize());
      n->init_leaf(parent, kNodeValues);
      return n;
    }
    node_type *new_leaf_root_node(const int max_count) {
      node_type *n = allocate(node_type::LeafSize(max_count));
      n->init_leaf(/*parent=*/n, max_count);
      return n;
    }
  
    // Deletion helper routines.
    void erase_same_node(iterator begin, iterator end);
    iterator erase_from_leaf_node(iterator begin, size_type to_erase);
    iterator rebalance_after_delete(iterator iter);
  
    // Deallocates a node of a certain size in bytes using the allocator.
    void deallocate(const size_type size, node_type *node) {
      absl::container_internal::Deallocate<node_type::Alignment()>(
          mutable_allocator(), node, size);
    }
  
    void delete_internal_node(node_type *node) {
      node->destroy(mutable_allocator());
      deallocate(node_type::InternalSize(), node);
    }
    void delete_leaf_node(node_type *node) {
      node->destroy(mutable_allocator());
      deallocate(node_type::LeafSize(node->max_count()), node);
    }
  
    // Rebalances or splits the node iter points to.
    void rebalance_or_split(iterator *iter);
  
    // Merges the values of left, right and the delimiting key on their parent
    // onto left, removing the delimiting key and deleting right.
    void merge_nodes(node_type *left, node_type *right);
  
    // Tries to merge node with its left or right sibling, and failing that,
    // rebalance with its left or right sibling. Returns true if a merge
    // occurred, at which point it is no longer valid to access node. Returns
    // false if no merging took place.
    bool try_merge_or_rebalance(iterator *iter);
  
    // Tries to shrink the height of the tree by 1.
    void try_shrink();
  
    iterator internal_end(iterator iter) {
      return iter.node != nullptr ? iter : end();
    }
    const_iterator internal_end(const_iterator iter) const {
      return iter.node != nullptr ? iter : end();
    }
  
    // Emplaces a value into the btree immediately before iter. Requires that
    // key(v) <= iter.key() and (--iter).key() <= key(v).
    template <typename... Args>
    iterator internal_emplace(iterator iter, Args &&... args);
  
    // Returns an iterator pointing to the first value >= the value "iter" is
    // pointing at. Note that "iter" might be pointing to an invalid location such
    // as iter.position == iter.node->finish(). This routine simply moves iter up
    // in the tree to a valid location.
    // Requires: iter.node is non-null.
    template <typename IterType>
    static IterType internal_last(IterType iter);
  
    // Returns an iterator pointing to the leaf position at which key would
    // reside in the tree. We provide 2 versions of internal_locate. The first
    // version uses a less-than comparator and is incapable of distinguishing when
    // there is an exact match. The second version is for the key-compare-to
    // specialization and distinguishes exact matches. The key-compare-to
    // specialization allows the caller to avoid a subsequent comparison to
    // determine if an exact match was made, which is important for keys with
    // expensive comparison, such as strings.
    template <typename K>
    SearchResult<iterator, is_key_compare_to::value> internal_locate(
        const K &key) const;
  
    template <typename K>
    SearchResult<iterator, false> internal_locate_impl(
        const K &key, std::false_type /* IsCompareTo */) const;
  
    template <typename K>
    SearchResult<iterator, true> internal_locate_impl(
        const K &key, std::true_type /* IsCompareTo */) const;
  
    // Internal routine which implements lower_bound().
    template <typename K>
    iterator internal_lower_bound(const K &key) const;
  
    // Internal routine which implements upper_bound().
    template <typename K>
    iterator internal_upper_bound(const K &key) const;
  
    // Internal routine which implements find().
    template <typename K>
    iterator internal_find(const K &key) const;
  
    // Deletes a node and all of its children.
    void internal_clear(node_type *node);
  
    // Verifies the tree structure of node.
    int internal_verify(const node_type *node, const key_type *lo,
                        const key_type *hi) const;
  
    node_stats internal_stats(const node_type *node) const {
      // The root can be a static empty node.
      if (node == nullptr || (node == root() && empty())) {
        return node_stats(0, 0);
      }
      if (node->leaf()) {
        return node_stats(1, 0);
      }
      node_stats res(0, 1);
      for (int i = node->start(); i <= node->finish(); ++i) {
        res += internal_stats(node->child(i));
      }
      return res;
    }
  
   public:
    // Exposed only for tests.
    static bool testonly_uses_linear_node_search() {
      return node_type::testonly_uses_linear_node_search();
    }
  
   private:
    // We use compressed tuple in order to save space because key_compare and
    // allocator_type are usually empty.
    absl::container_internal::CompressedTuple<key_compare, allocator_type,
                                              node_type *>
        root_;
  
    // A pointer to the rightmost node. Note that the leftmost node is stored as
    // the root's parent.
    node_type *rightmost_;
  
    // Number of values.
    size_type size_;
  };
  
  ////
  // btree_node methods
  template <typename P>
  template <typename... Args>
  inline void btree_node<P>::emplace_value(const size_type i,
                                           allocator_type *alloc,
                                           Args &&... args) {
    assert(i >= start());
    assert(i <= finish());
    // Shift old values to create space for new value and then construct it in
    // place.
    if (i < finish()) {
      transfer_n_backward(finish() - i, /*dest_i=*/i + 1, /*src_i=*/i, this,
                          alloc);
    }
    value_init(i, alloc, std::forward<Args>(args)...);
    set_finish(finish() + 1);
  
    if (!leaf() && finish() > i + 1) {
      for (int j = finish(); j > i + 1; --j) {
        set_child(j, child(j - 1));
      }
      clear_child(i + 1);
    }
  }
  
  template <typename P>
  inline void btree_node<P>::remove_value(const int i, allocator_type *alloc) {
    if (!leaf() && finish() > i + 1) {
      assert(child(i + 1)->count() == 0);
      for (size_type j = i + 1; j < finish(); ++j) {
        set_child(j, child(j + 1));
      }
      clear_child(finish());
    }
  
    remove_values_ignore_children(i, /*to_erase=*/1, alloc);
  }
  
  template <typename P>
  inline void btree_node<P>::remove_values_ignore_children(
      const int i, const int to_erase, allocator_type *alloc) {
    params_type::move(alloc, slot(i + to_erase), finish_slot(), slot(i));
    for (int j = finish() - to_erase; j < finish(); ++j) {
      value_destroy(j, alloc);
    }
    set_finish(finish() - to_erase);
  }
  
  template <typename P>
  void btree_node<P>::rebalance_right_to_left(const int to_move,
                                              btree_node *right,
                                              allocator_type *alloc) {
    assert(parent() == right->parent());
    assert(position() + 1 == right->position());
    assert(right->count() >= count());
    assert(to_move >= 1);
    assert(to_move <= right->count());
  
    // 1) Move the delimiting value in the parent to the left node.
    transfer(finish(), position(), parent(), alloc);
  
    // 2) Move the (to_move - 1) values from the right node to the left node.
    transfer_n(to_move - 1, finish() + 1, right->start(), right, alloc);
  
    // 3) Move the new delimiting value to the parent from the right node.
    parent()->transfer(position(), right->start() + to_move - 1, right, alloc);
  
    // 4) Shift the values in the right node to their correct positions.
    right->transfer_n(right->count() - to_move, right->start(),
                      right->start() + to_move, right, alloc);
  
    if (!leaf()) {
      // Move the child pointers from the right to the left node.
      for (int i = 0; i < to_move; ++i) {
        init_child(finish() + i + 1, right->child(i));
      }
      for (int i = right->start(); i <= right->finish() - to_move; ++i) {
        assert(i + to_move <= right->max_count());
        right->init_child(i, right->child(i + to_move));
        right->clear_child(i + to_move);
      }
    }
  
    // Fixup `finish` on the left and right nodes.
    set_finish(finish() + to_move);
    right->set_finish(right->finish() - to_move);
  }
  
  template <typename P>
  void btree_node<P>::rebalance_left_to_right(const int to_move,
                                              btree_node *right,
                                              allocator_type *alloc) {
    assert(parent() == right->parent());
    assert(position() + 1 == right->position());
    assert(count() >= right->count());
    assert(to_move >= 1);
    assert(to_move <= count());
  
    // Values in the right node are shifted to the right to make room for the
    // new to_move values. Then, the delimiting value in the parent and the
    // other (to_move - 1) values in the left node are moved into the right node.
    // Lastly, a new delimiting value is moved from the left node into the
    // parent, and the remaining empty left node entries are destroyed.
  
    // 1) Shift existing values in the right node to their correct positions.
    right->transfer_n_backward(right->count(), right->start() + to_move,
                               right->start(), right, alloc);
  
    // 2) Move the delimiting value in the parent to the right node.
    right->transfer(right->start() + to_move - 1, position(), parent(), alloc);
  
    // 3) Move the (to_move - 1) values from the left node to the right node.
    right->transfer_n(to_move - 1, right->start(), finish() - (to_move - 1), this,
                      alloc);
  
    // 4) Move the new delimiting value to the parent from the left node.
    parent()->transfer(position(), finish() - to_move, this, alloc);
  
    if (!leaf()) {
      // Move the child pointers from the left to the right node.
      for (int i = right->finish(); i >= right->start(); --i) {
        right->init_child(i + to_move, right->child(i));
        right->clear_child(i);
      }
      for (int i = 1; i <= to_move; ++i) {
        right->init_child(i - 1, child(finish() - to_move + i));
        clear_child(finish() - to_move + i);
      }
    }
  
    // Fixup the counts on the left and right nodes.
    set_finish(finish() - to_move);
    right->set_finish(right->finish() + to_move);
  }
  
  template <typename P>
  void btree_node<P>::split(const int insert_position, btree_node *dest,
                            allocator_type *alloc) {
    assert(dest->count() == 0);
    assert(max_count() == kNodeValues);
  
    // We bias the split based on the position being inserted. If we're
    // inserting at the beginning of the left node then bias the split to put
    // more values on the right node. If we're inserting at the end of the
    // right node then bias the split to put more values on the left node.
    if (insert_position == start()) {
      dest->set_finish(dest->start() + finish() - 1);
    } else if (insert_position == kNodeValues) {
      dest->set_finish(dest->start());
    } else {
      dest->set_finish(dest->start() + count() / 2);
    }
    set_finish(finish() - dest->count());
    assert(count() >= 1);
  
    // Move values from the left sibling to the right sibling.
    dest->transfer_n(dest->count(), dest->start(), finish(), this, alloc);
  
    // The split key is the largest value in the left sibling.
    --mutable_finish();
    parent()->emplace_value(position(), alloc, finish_slot());
    value_destroy(finish(), alloc);
    parent()->init_child(position() + 1, dest);
  
    if (!leaf()) {
      for (int i = dest->start(), j = finish() + 1; i <= dest->finish();
           ++i, ++j) {
        assert(child(j) != nullptr);
        dest->init_child(i, child(j));
        clear_child(j);
      }
    }
  }
  
  template <typename P>
  void btree_node<P>::merge(btree_node *src, allocator_type *alloc) {
    assert(parent() == src->parent());
    assert(position() + 1 == src->position());
  
    // Move the delimiting value to the left node.
    value_init(finish(), alloc, parent()->slot(position()));
  
    // Move the values from the right to the left node.
    transfer_n(src->count(), finish() + 1, src->start(), src, alloc);
  
    if (!leaf()) {
      // Move the child pointers from the right to the left node.
      for (int i = src->start(), j = finish() + 1; i <= src->finish(); ++i, ++j) {
        init_child(j, src->child(i));
        src->clear_child(i);
      }
    }
  
    // Fixup `finish` on the src and dest nodes.
    set_finish(start() + 1 + count() + src->count());
    src->set_finish(src->start());
  
    // Remove the value on the parent node.
    parent()->remove_value(position(), alloc);
  }
  
  ////
  // btree_iterator methods
  template <typename N, typename R, typename P>
  void btree_iterator<N, R, P>::increment_slow() {
    if (node->leaf()) {
      assert(position >= node->finish());
      btree_iterator save(*this);
      while (position == node->finish() && !node->is_root()) {
        assert(node->parent()->child(node->position()) == node);
        position = node->position();
        node = node->parent();
      }
      // TODO(ezb): assert we aren't incrementing end() instead of handling.
      if (position == node->finish()) {
        *this = save;
      }
    } else {
      assert(position < node->finish());
      node = node->child(position + 1);
      while (!node->leaf()) {
        node = node->start_child();
      }
      position = node->start();
    }
  }
  
  template <typename N, typename R, typename P>
  void btree_iterator<N, R, P>::decrement_slow() {
    if (node->leaf()) {
      assert(position <= -1);
      btree_iterator save(*this);
      while (position < node->start() && !node->is_root()) {
        assert(node->parent()->child(node->position()) == node);
        position = node->position() - 1;
        node = node->parent();
      }
      // TODO(ezb): assert we aren't decrementing begin() instead of handling.
      if (position < node->start()) {
        *this = save;
      }
    } else {
      assert(position >= node->start());
      node = node->child(position);
      while (!node->leaf()) {
        node = node->child(node->finish());
      }
      position = node->finish() - 1;
    }
  }
  
  ////
  // btree methods
  template <typename P>
  template <typename Btree>
  void btree<P>::copy_or_move_values_in_order(Btree *other) {
    static_assert(std::is_same<btree, Btree>::value ||
                      std::is_same<const btree, Btree>::value,
                  "Btree type must be same or const.");
    assert(empty());
  
    // We can avoid key comparisons because we know the order of the
    // values is the same order we'll store them in.
    auto iter = other->begin();
    if (iter == other->end()) return;
    insert_multi(maybe_move_from_iterator(iter));
    ++iter;
    for (; iter != other->end(); ++iter) {
      // If the btree is not empty, we can just insert the new value at the end
      // of the tree.
      internal_emplace(end(), maybe_move_from_iterator(iter));
    }
  }
  
  template <typename P>
  constexpr bool btree<P>::static_assert_validation() {
    static_assert(std::is_nothrow_copy_constructible<key_compare>::value,
                  "Key comparison must be nothrow copy constructible");
    static_assert(std::is_nothrow_copy_constructible<allocator_type>::value,
                  "Allocator must be nothrow copy constructible");
    static_assert(type_traits_internal::is_trivially_copyable<iterator>::value,
                  "iterator not trivially copyable.");
  
    // Note: We assert that kTargetValues, which is computed from
    // Params::kTargetNodeSize, must fit the node_type::field_type.
    static_assert(
        kNodeValues < (1 << (8 * sizeof(typename node_type::field_type))),
        "target node size too large");
  
    // Verify that key_compare returns an absl::{weak,strong}_ordering or bool.
  #if !defined(__cpp_lib_is_invocable)
    using compare_result_type =
        absl::result_of_t<key_compare(key_type, key_type)>;
  #else
    using compare_result_type =
        absl::invoke_result_t<key_compare, key_type, key_type>;
  #endif
    static_assert(
        std::is_same<compare_result_type, bool>::value ||
            std::is_convertible<compare_result_type, absl::weak_ordering>::value,
        "key comparison function must return absl::{weak,strong}_ordering or "
        "bool.");
  
    // Test the assumption made in setting kNodeValueSpace.
    static_assert(node_type::MinimumOverhead() >= sizeof(void *) + 4,
                  "node space assumption incorrect");
  
    return true;
  }
  
  template <typename P>
  btree<P>::btree(const key_compare &comp, const allocator_type &alloc)
      : root_(comp, alloc, EmptyNode()), rightmost_(EmptyNode()), size_(0) {}
  
  template <typename P>
  btree<P>::btree(const btree &other)
      : btree(other.key_comp(), other.allocator()) {
    copy_or_move_values_in_order(&other);
  }
  
  template <typename P>
  template <typename... Args>
  auto btree<P>::insert_unique(const key_type &key, Args &&... args)
      -> std::pair<iterator, bool> {
    if (empty()) {
      mutable_root() = rightmost_ = new_leaf_root_node(1);
    }
  
    auto res = internal_locate(key);
    iterator &iter = res.value;
  
    if (res.HasMatch()) {
      if (res.IsEq()) {
        // The key already exists in the tree, do nothing.
        return {iter, false};
      }
    } else {
      iterator last = internal_last(iter);
      if (last.node && !compare_keys(key, last.key())) {
        // The key already exists in the tree, do nothing.
        return {last, false};
      }
    }
    return {internal_emplace(iter, std::forward<Args>(args)...), true};
  }
  
  template <typename P>
  template <typename... Args>
  inline auto btree<P>::insert_hint_unique(iterator position, const key_type &key,
                                           Args &&... args)
      -> std::pair<iterator, bool> {
    if (!empty()) {
      if (position == end() || compare_keys(key, position.key())) {
        if (position == begin() || compare_keys(std::prev(position).key(), key)) {
          // prev.key() < key < position.key()
          return {internal_emplace(position, std::forward<Args>(args)...), true};
        }
      } else if (compare_keys(position.key(), key)) {
        ++position;
        if (position == end() || compare_keys(key, position.key())) {
          // {original `position`}.key() < key < {current `position`}.key()
          return {internal_emplace(position, std::forward<Args>(args)...), true};
        }
      } else {
        // position.key() == key
        return {position, false};
      }
    }
    return insert_unique(key, std::forward<Args>(args)...);
  }
  
  template <typename P>
  template <typename InputIterator>
  void btree<P>::insert_iterator_unique(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
      insert_hint_unique(end(), params_type::key(*b), *b);
    }
  }
  
  template <typename P>
  template <typename ValueType>
  auto btree<P>::insert_multi(const key_type &key, ValueType &&v) -> iterator {
    if (empty()) {
      mutable_root() = rightmost_ = new_leaf_root_node(1);
    }
  
    iterator iter = internal_upper_bound(key);
    if (iter.node == nullptr) {
      iter = end();
    }
    return internal_emplace(iter, std::forward<ValueType>(v));
  }
  
  template <typename P>
  template <typename ValueType>
  auto btree<P>::insert_hint_multi(iterator position, ValueType &&v) -> iterator {
    if (!empty()) {
      const key_type &key = params_type::key(v);
      if (position == end() || !compare_keys(position.key(), key)) {
        if (position == begin() ||
            !compare_keys(key, std::prev(position).key())) {
          // prev.key() <= key <= position.key()
          return internal_emplace(position, std::forward<ValueType>(v));
        }
      } else {
        ++position;
        if (position == end() || !compare_keys(position.key(), key)) {
          // {original `position`}.key() < key < {current `position`}.key()
          return internal_emplace(position, std::forward<ValueType>(v));
        }
      }
    }
    return insert_multi(std::forward<ValueType>(v));
  }
  
  template <typename P>
  template <typename InputIterator>
  void btree<P>::insert_iterator_multi(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
      insert_hint_multi(end(), *b);
    }
  }
  
  template <typename P>
  auto btree<P>::operator=(const btree &other) -> btree & {
    if (this != &other) {
      clear();
  
      *mutable_key_comp() = other.key_comp();
      if (absl::allocator_traits<
              allocator_type>::propagate_on_container_copy_assignment::value) {
        *mutable_allocator() = other.allocator();
      }
  
      copy_or_move_values_in_order(&other);
    }
    return *this;
  }
  
  template <typename P>
  auto btree<P>::operator=(btree &&other) noexcept -> btree & {
    if (this != &other) {
      clear();
  
      using std::swap;
      if (absl::allocator_traits<
              allocator_type>::propagate_on_container_copy_assignment::value) {
        // Note: `root_` also contains the allocator and the key comparator.
        swap(root_, other.root_);
        swap(rightmost_, other.rightmost_);
        swap(size_, other.size_);
      } else {
        if (allocator() == other.allocator()) {
          swap(mutable_root(), other.mutable_root());
          swap(*mutable_key_comp(), *other.mutable_key_comp());
          swap(rightmost_, other.rightmost_);
          swap(size_, other.size_);
        } else {
          // We aren't allowed to propagate the allocator and the allocator is
          // different so we can't take over its memory. We must move each element
          // individually. We need both `other` and `this` to have `other`s key
          // comparator while moving the values so we can't swap the key
          // comparators.
          *mutable_key_comp() = other.key_comp();
          copy_or_move_values_in_order(&other);
        }
      }
    }
    return *this;
  }
  
  template <typename P>
  auto btree<P>::erase(iterator iter) -> iterator {
    bool internal_delete = false;
    if (!iter.node->leaf()) {
      // Deletion of a value on an internal node. First, move the largest value
      // from our left child here, then delete that position (in remove_value()
      // below). We can get to the largest value from our left child by
      // decrementing iter.
      iterator internal_iter(iter);
      --iter;
      assert(iter.node->leaf());
      params_type::move(mutable_allocator(), iter.node->slot(iter.position),
                        internal_iter.node->slot(internal_iter.position));
      internal_delete = true;
    }
  
    // Delete the key from the leaf.
    iter.node->remove_value(iter.position, mutable_allocator());
    --size_;
  
    // We want to return the next value after the one we just erased. If we
    // erased from an internal node (internal_delete == true), then the next
    // value is ++(++iter). If we erased from a leaf node (internal_delete ==
    // false) then the next value is ++iter. Note that ++iter may point to an
    // internal node and the value in the internal node may move to a leaf node
    // (iter.node) when rebalancing is performed at the leaf level.
  
    iterator res = rebalance_after_delete(iter);
  
    // If we erased from an internal node, advance the iterator.
    if (internal_delete) {
      ++res;
    }
    return res;
  }
  
  template <typename P>
  auto btree<P>::rebalance_after_delete(iterator iter) -> iterator {
    // Merge/rebalance as we walk back up the tree.
    iterator res(iter);
    bool first_iteration = true;
    for (;;) {
      if (iter.node == root()) {
        try_shrink();
        if (empty()) {
          return end();
        }
        break;
      }
      if (iter.node->count() >= kMinNodeValues) {
        break;
      }
      bool merged = try_merge_or_rebalance(&iter);
      // On the first iteration, we should update `res` with `iter` because `res`
      // may have been invalidated.
      if (first_iteration) {
        res = iter;
        first_iteration = false;
      }
      if (!merged) {
        break;
      }
      iter.position = iter.node->position();
      iter.node = iter.node->parent();
    }
  
    // Adjust our return value. If we're pointing at the end of a node, advance
    // the iterator.
    if (res.position == res.node->finish()) {
      res.position = res.node->finish() - 1;
      ++res;
    }
  
    return res;
  }
  
  template <typename P>
  auto btree<P>::erase_range(iterator begin, iterator end)
      -> std::pair<size_type, iterator> {
    difference_type count = std::distance(begin, end);
    assert(count >= 0);
  
    if (count == 0) {
      return {0, begin};
    }
  
    if (count == size_) {
      clear();
      return {count, this->end()};
    }
  
    if (begin.node == end.node) {
      erase_same_node(begin, end);
      size_ -= count;
      return {count, rebalance_after_delete(begin)};
    }
  
    const size_type target_size = size_ - count;
    while (size_ > target_size) {
      if (begin.node->leaf()) {
        const size_type remaining_to_erase = size_ - target_size;
        const size_type remaining_in_node = begin.node->finish() - begin.position;
        begin = erase_from_leaf_node(
            begin, (std::min)(remaining_to_erase, remaining_in_node));
      } else {
        begin = erase(begin);
      }
    }
    return {count, begin};
  }
  
  template <typename P>
  void btree<P>::erase_same_node(iterator begin, iterator end) {
    assert(begin.node == end.node);
    assert(end.position > begin.position);
  
    node_type *node = begin.node;
    size_type to_erase = end.position - begin.position;
    if (!node->leaf()) {
      // Delete all children between begin and end.
      for (size_type i = 0; i < to_erase; ++i) {
        internal_clear(node->child(begin.position + i + 1));
      }
      // Rotate children after end into new positions.
      for (size_type i = begin.position + to_erase + 1; i <= node->finish();
           ++i) {
        node->set_child(i - to_erase, node->child(i));
        node->clear_child(i);
      }
    }
    node->remove_values_ignore_children(begin.position, to_erase,
                                        mutable_allocator());
  
    // Do not need to update rightmost_, because
    // * either end == this->end(), and therefore node == rightmost_, and still
    //   exists
    // * or end != this->end(), and therefore rightmost_ hasn't been erased, since
    //   it wasn't covered in [begin, end)
  }
  
  template <typename P>
  auto btree<P>::erase_from_leaf_node(iterator begin, size_type to_erase)
      -> iterator {
    node_type *node = begin.node;
    assert(node->leaf());
    assert(node->finish() > begin.position);
    assert(begin.position + to_erase <= node->finish());
  
    node->remove_values_ignore_children(begin.position, to_erase,
                                        mutable_allocator());
  
    size_ -= to_erase;
  
    return rebalance_after_delete(begin);
  }
  
  template <typename P>
  template <typename K>
  auto btree<P>::erase_unique(const K &key) -> size_type {
    const iterator iter = internal_find(key);
    if (iter.node == nullptr) {
      // The key doesn't exist in the tree, return nothing done.
      return 0;
    }
    erase(iter);
    return 1;
  }
  
  template <typename P>
  template <typename K>
  auto btree<P>::erase_multi(const K &key) -> size_type {
    const iterator begin = internal_lower_bound(key);
    if (begin.node == nullptr) {
      // The key doesn't exist in the tree, return nothing done.
      return 0;
    }
    // Delete all of the keys between begin and upper_bound(key).
    const iterator end = internal_end(internal_upper_bound(key));
    return erase_range(begin, end).first;
  }
  
  template <typename P>
  void btree<P>::clear() {
    if (!empty()) {
      internal_clear(root());
    }
    mutable_root() = EmptyNode();
    rightmost_ = EmptyNode();
    size_ = 0;
  }
  
  template <typename P>
  void btree<P>::swap(btree &other) {
    using std::swap;
    if (absl::allocator_traits<
            allocator_type>::propagate_on_container_swap::value) {
      // Note: `root_` also contains the allocator and the key comparator.
      swap(root_, other.root_);
    } else {
      // It's undefined behavior if the allocators are unequal here.
      assert(allocator() == other.allocator());
      swap(mutable_root(), other.mutable_root());
      swap(*mutable_key_comp(), *other.mutable_key_comp());
    }
    swap(rightmost_, other.rightmost_);
    swap(size_, other.size_);
  }
  
  template <typename P>
  void btree<P>::verify() const {
    assert(root() != nullptr);
    assert(leftmost() != nullptr);
    assert(rightmost_ != nullptr);
    assert(empty() || size() == internal_verify(root(), nullptr, nullptr));
    assert(leftmost() == (++const_iterator(root(), -1)).node);
    assert(rightmost_ == (--const_iterator(root(), root()->finish())).node);
    assert(leftmost()->leaf());
    assert(rightmost_->leaf());
  }
  
  template <typename P>
  void btree<P>::rebalance_or_split(iterator *iter) {
    node_type *&node = iter->node;
    int &insert_position = iter->position;
    assert(node->count() == node->max_count());
    assert(kNodeValues == node->max_count());
  
    // First try to make room on the node by rebalancing.
    node_type *parent = node->parent();
    if (node != root()) {
      if (node->position() > parent->start()) {
        // Try rebalancing with our left sibling.
        node_type *left = parent->child(node->position() - 1);
        assert(left->max_count() == kNodeValues);
        if (left->count() < kNodeValues) {
          // We bias rebalancing based on the position being inserted. If we're
          // inserting at the end of the right node then we bias rebalancing to
          // fill up the left node.
          int to_move = (kNodeValues - left->count()) /
                        (1 + (insert_position < kNodeValues));
          to_move = (std::max)(1, to_move);
  
          if (insert_position - to_move >= node->start() ||
              left->count() + to_move < kNodeValues) {
            left->rebalance_right_to_left(to_move, node, mutable_allocator());
  
            assert(node->max_count() - node->count() == to_move);
            insert_position = insert_position - to_move;
            if (insert_position < node->start()) {
              insert_position = insert_position + left->count() + 1;
              node = left;
            }
  
            assert(node->count() < node->max_count());
            return;
          }
        }
      }
  
      if (node->position() < parent->finish()) {
        // Try rebalancing with our right sibling.
        node_type *right = parent->child(node->position() + 1);
        assert(right->max_count() == kNodeValues);
        if (right->count() < kNodeValues) {
          // We bias rebalancing based on the position being inserted. If we're
          // inserting at the beginning of the left node then we bias rebalancing
          // to fill up the right node.
          int to_move = (kNodeValues - right->count()) /
                        (1 + (insert_position > node->start()));
          to_move = (std::max)(1, to_move);
  
          if (insert_position <= node->finish() - to_move ||
              right->count() + to_move < kNodeValues) {
            node->rebalance_left_to_right(to_move, right, mutable_allocator());
  
            if (insert_position > node->finish()) {
              insert_position = insert_position - node->count() - 1;
              node = right;
            }
  
            assert(node->count() < node->max_count());
            return;
          }
        }
      }
  
      // Rebalancing failed, make sure there is room on the parent node for a new
      // value.
      assert(parent->max_count() == kNodeValues);
      if (parent->count() == kNodeValues) {
        iterator parent_iter(node->parent(), node->position());
        rebalance_or_split(&parent_iter);
      }
    } else {
      // Rebalancing not possible because this is the root node.
      // Create a new root node and set the current root node as the child of the
      // new root.
      parent = new_internal_node(parent);
      parent->init_child(parent->start(), root());
      mutable_root() = parent;
      // If the former root was a leaf node, then it's now the rightmost node.
      assert(!parent->start_child()->leaf() ||
             parent->start_child() == rightmost_);
    }
  
    // Split the node.
    node_type *split_node;
    if (node->leaf()) {
      split_node = new_leaf_node(parent);
      node->split(insert_position, split_node, mutable_allocator());
      if (rightmost_ == node) rightmost_ = split_node;
    } else {
      split_node = new_internal_node(parent);
      node->split(insert_position, split_node, mutable_allocator());
    }
  
    if (insert_position > node->finish()) {
      insert_position = insert_position - node->count() - 1;
      node = split_node;
    }
  }
  
  template <typename P>
  void btree<P>::merge_nodes(node_type *left, node_type *right) {
    left->merge(right, mutable_allocator());
    if (right->leaf()) {
      if (rightmost_ == right) rightmost_ = left;
      delete_leaf_node(right);
    } else {
      delete_internal_node(right);
    }
  }
  
  template <typename P>
  bool btree<P>::try_merge_or_rebalance(iterator *iter) {
    node_type *parent = iter->node->parent();
    if (iter->node->position() > parent->start()) {
      // Try merging with our left sibling.
      node_type *left = parent->child(iter->node->position() - 1);
      assert(left->max_count() == kNodeValues);
      if (1 + left->count() + iter->node->count() <= kNodeValues) {
        iter->position += 1 + left->count();
        merge_nodes(left, iter->node);
        iter->node = left;
        return true;
      }
    }
    if (iter->node->position() < parent->finish()) {
      // Try merging with our right sibling.
      node_type *right = parent->child(iter->node->position() + 1);
      assert(right->max_count() == kNodeValues);
      if (1 + iter->node->count() + right->count() <= kNodeValues) {
        merge_nodes(iter->node, right);
        return true;
      }
      // Try rebalancing with our right sibling. We don't perform rebalancing if
      // we deleted the first element from iter->node and the node is not
      // empty. This is a small optimization for the common pattern of deleting
      // from the front of the tree.
      if (right->count() > kMinNodeValues &&
          (iter->node->count() == 0 || iter->position > iter->node->start())) {
        int to_move = (right->count() - iter->node->count()) / 2;
        to_move = (std::min)(to_move, right->count() - 1);
        iter->node->rebalance_right_to_left(to_move, right, mutable_allocator());
        return false;
      }
    }
    if (iter->node->position() > parent->start()) {
      // Try rebalancing with our left sibling. We don't perform rebalancing if
      // we deleted the last element from iter->node and the node is not
      // empty. This is a small optimization for the common pattern of deleting
      // from the back of the tree.
      node_type *left = parent->child(iter->node->position() - 1);
      if (left->count() > kMinNodeValues &&
          (iter->node->count() == 0 || iter->position < iter->node->finish())) {
        int to_move = (left->count() - iter->node->count()) / 2;
        to_move = (std::min)(to_move, left->count() - 1);
        left->rebalance_left_to_right(to_move, iter->node, mutable_allocator());
        iter->position += to_move;
        return false;
      }
    }
    return false;
  }
  
  template <typename P>
  void btree<P>::try_shrink() {
    if (root()->count() > 0) {
      return;
    }
    // Deleted the last item on the root node, shrink the height of the tree.
    if (root()->leaf()) {
      assert(size() == 0);
      delete_leaf_node(root());
      mutable_root() = rightmost_ = EmptyNode();
    } else {
      node_type *child = root()->start_child();
      child->make_root();
      delete_internal_node(root());
      mutable_root() = child;
    }
  }
  
  template <typename P>
  template <typename IterType>
  inline IterType btree<P>::internal_last(IterType iter) {
    assert(iter.node != nullptr);
    while (iter.position == iter.node->finish()) {
      iter.position = iter.node->position();
      iter.node = iter.node->parent();
      if (iter.node->leaf()) {
        iter.node = nullptr;
        break;
      }
    }
    return iter;
  }
  
  template <typename P>
  template <typename... Args>
  inline auto btree<P>::internal_emplace(iterator iter, Args &&... args)
      -> iterator {
    if (!iter.node->leaf()) {
      // We can't insert on an internal node. Instead, we'll insert after the
      // previous value which is guaranteed to be on a leaf node.
      --iter;
      ++iter.position;
    }
    const int max_count = iter.node->max_count();
    allocator_type *alloc = mutable_allocator();
    if (iter.node->count() == max_count) {
      // Make room in the leaf for the new item.
      if (max_count < kNodeValues) {
        // Insertion into the root where the root is smaller than the full node
        // size. Simply grow the size of the root node.
        assert(iter.node == root());
        iter.node =
            new_leaf_root_node((std::min<int>)(kNodeValues, 2 * max_count));
        // Transfer the values from the old root to the new root.
        node_type *old_root = root();
        node_type *new_root = iter.node;
        new_root->transfer_n(old_root->count(), new_root->start(),
                             old_root->start(), old_root, alloc);
        new_root->set_finish(old_root->finish());
        old_root->set_finish(old_root->start());
        delete_leaf_node(old_root);
        mutable_root() = rightmost_ = new_root;
      } else {
        rebalance_or_split(&iter);
      }
    }
    iter.node->emplace_value(iter.position, alloc, std::forward<Args>(args)...);
    ++size_;
    return iter;
  }
  
  template <typename P>
  template <typename K>
  inline auto btree<P>::internal_locate(const K &key) const
      -> SearchResult<iterator, is_key_compare_to::value> {
    return internal_locate_impl(key, is_key_compare_to());
  }
  
  template <typename P>
  template <typename K>
  inline auto btree<P>::internal_locate_impl(
      const K &key, std::false_type /* IsCompareTo */) const
      -> SearchResult<iterator, false> {
    iterator iter(const_cast<node_type *>(root()));
    for (;;) {
      iter.position = iter.node->lower_bound(key, key_comp()).value;
      // NOTE: we don't need to walk all the way down the tree if the keys are
      // equal, but determining equality would require doing an extra comparison
      // on each node on the way down, and we will need to go all the way to the
      // leaf node in the expected case.
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return {iter};
  }
  
  template <typename P>
  template <typename K>
  inline auto btree<P>::internal_locate_impl(
      const K &key, std::true_type /* IsCompareTo */) const
      -> SearchResult<iterator, true> {
    iterator iter(const_cast<node_type *>(root()));
    for (;;) {
      SearchResult<int, true> res = iter.node->lower_bound(key, key_comp());
      iter.position = res.value;
      if (res.match == MatchKind::kEq) {
        return {iter, MatchKind::kEq};
      }
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return {iter, MatchKind::kNe};
  }
  
  template <typename P>
  template <typename K>
  auto btree<P>::internal_lower_bound(const K &key) const -> iterator {
    iterator iter(const_cast<node_type *>(root()));
    for (;;) {
      iter.position = iter.node->lower_bound(key, key_comp()).value;
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return internal_last(iter);
  }
  
  template <typename P>
  template <typename K>
  auto btree<P>::internal_upper_bound(const K &key) const -> iterator {
    iterator iter(const_cast<node_type *>(root()));
    for (;;) {
      iter.position = iter.node->upper_bound(key, key_comp());
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return internal_last(iter);
  }
  
  template <typename P>
  template <typename K>
  auto btree<P>::internal_find(const K &key) const -> iterator {
    auto res = internal_locate(key);
    if (res.HasMatch()) {
      if (res.IsEq()) {
        return res.value;
      }
    } else {
      const iterator iter = internal_last(res.value);
      if (iter.node != nullptr && !compare_keys(key, iter.key())) {
        return iter;
      }
    }
    return {nullptr, 0};
  }
  
  template <typename P>
  void btree<P>::internal_clear(node_type *node) {
    if (!node->leaf()) {
      for (int i = node->start(); i <= node->finish(); ++i) {
        internal_clear(node->child(i));
      }
      delete_internal_node(node);
    } else {
      delete_leaf_node(node);
    }
  }
  
  template <typename P>
  int btree<P>::internal_verify(const node_type *node, const key_type *lo,
                                const key_type *hi) const {
    assert(node->count() > 0);
    assert(node->count() <= node->max_count());
    if (lo) {
      assert(!compare_keys(node->key(node->start()), *lo));
    }
    if (hi) {
      assert(!compare_keys(*hi, node->key(node->finish() - 1)));
    }
    for (int i = node->start() + 1; i < node->finish(); ++i) {
      assert(!compare_keys(node->key(i), node->key(i - 1)));
    }
    int count = node->count();
    if (!node->leaf()) {
      for (int i = node->start(); i <= node->finish(); ++i) {
        assert(node->child(i) != nullptr);
        assert(node->child(i)->parent() == node);
        assert(node->child(i)->position() == i);
        count += internal_verify(node->child(i),
                                 i == node->start() ? lo : &node->key(i - 1),
                                 i == node->finish() ? hi : &node->key(i));
      }
    }
    return count;
  }
  
  }  // namespace container_internal
  ABSL_NAMESPACE_END
  }  // namespace absl
  
  #endif  // ABSL_CONTAINER_INTERNAL_BTREE_H_
  