text: // Copyright 2013 Google Inc. All Rights Reserved.
  //
  // 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
  //
  //     http://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 both
  // smaller and faster than STL set/map. 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> consumes 20
  // bytes for each value stored. 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> may use much
  // less memory per stored value. For the random insertion benchmark in
  // btree_test.cc, a btree_set<int32> with node-size of 256 uses 4.9 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. 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).  See also safe_btree.
  
  // PERFORMANCE
  //
  //   btree_bench --benchmarks=. 2>&1 | ./benchmarks.awk
  //
  // Run on pmattis-warp.nyc (4 X 2200 MHz CPUs); 2010/03/04-15:23:06
  // Benchmark                 STL(ns) B-Tree(ns) @    <size>
  // --------------------------------------------------------
  // BM_set_int32_insert        1516      608  +59.89%  <256>    [40.0,  5.2]
  // BM_set_int32_lookup        1160      414  +64.31%  <256>    [40.0,  5.2]
  // BM_set_int32_fulllookup     960      410  +57.29%  <256>    [40.0,  4.4]
  // BM_set_int32_delete        1741      528  +69.67%  <256>    [40.0,  5.2]
  // BM_set_int32_queueaddrem   3078     1046  +66.02%  <256>    [40.0,  5.5]
  // BM_set_int32_mixedaddrem   3600     1384  +61.56%  <256>    [40.0,  5.3]
  // BM_set_int32_fifo           227      113  +50.22%  <256>    [40.0,  4.4]
  // BM_set_int32_fwditer        158       26  +83.54%  <256>    [40.0,  5.2]
  // BM_map_int32_insert        1551      636  +58.99%  <256>    [48.0, 10.5]
  // BM_map_int32_lookup        1200      508  +57.67%  <256>    [48.0, 10.5]
  // BM_map_int32_fulllookup     989      487  +50.76%  <256>    [48.0,  8.8]
  // BM_map_int32_delete        1794      628  +64.99%  <256>    [48.0, 10.5]
  // BM_map_int32_queueaddrem   3189     1266  +60.30%  <256>    [48.0, 11.6]
  // BM_map_int32_mixedaddrem   3822     1623  +57.54%  <256>    [48.0, 10.9]
  // BM_map_int32_fifo           151      134  +11.26%  <256>    [48.0,  8.8]
  // BM_map_int32_fwditer        161       32  +80.12%  <256>    [48.0, 10.5]
  // BM_set_int64_insert        1546      636  +58.86%  <256>    [40.0, 10.5]
  // BM_set_int64_lookup        1200      512  +57.33%  <256>    [40.0, 10.5]
  // BM_set_int64_fulllookup     971      487  +49.85%  <256>    [40.0,  8.8]
  // BM_set_int64_delete        1745      616  +64.70%  <256>    [40.0, 10.5]
  // BM_set_int64_queueaddrem   3163     1195  +62.22%  <256>    [40.0, 11.6]
  // BM_set_int64_mixedaddrem   3760     1564  +58.40%  <256>    [40.0, 10.9]
  // BM_set_int64_fifo           146      103  +29.45%  <256>    [40.0,  8.8]
  // BM_set_int64_fwditer        162       31  +80.86%  <256>    [40.0, 10.5]
  // BM_map_int64_insert        1551      720  +53.58%  <256>    [48.0, 20.7]
  // BM_map_int64_lookup        1214      612  +49.59%  <256>    [48.0, 20.7]
  // BM_map_int64_fulllookup     994      592  +40.44%  <256>    [48.0, 17.2]
  // BM_map_int64_delete        1778      764  +57.03%  <256>    [48.0, 20.7]
  // BM_map_int64_queueaddrem   3189     1547  +51.49%  <256>    [48.0, 20.9]
  // BM_map_int64_mixedaddrem   3779     1887  +50.07%  <256>    [48.0, 21.6]
  // BM_map_int64_fifo           147      145   +1.36%  <256>    [48.0, 17.2]
  // BM_map_int64_fwditer        162       41  +74.69%  <256>    [48.0, 20.7]
  // BM_set_string_insert       1989     1966   +1.16%  <256>    [64.0, 44.5]
  // BM_set_string_lookup       1709     1600   +6.38%  <256>    [64.0, 44.5]
  // BM_set_string_fulllookup   1573     1529   +2.80%  <256>    [64.0, 35.4]
  // BM_set_string_delete       2520     1920  +23.81%  <256>    [64.0, 44.5]
  // BM_set_string_queueaddrem  4706     4309   +8.44%  <256>    [64.0, 48.3]
  // BM_set_string_mixedaddrem  5080     4654   +8.39%  <256>    [64.0, 46.7]
  // BM_set_string_fifo          318      512  -61.01%  <256>    [64.0, 35.4]
  // BM_set_string_fwditer       182       93  +48.90%  <256>    [64.0, 44.5]
  // BM_map_string_insert       2600     2227  +14.35%  <256>    [72.0, 55.8]
  // BM_map_string_lookup       2068     1730  +16.34%  <256>    [72.0, 55.8]
  // BM_map_string_fulllookup   1859     1618  +12.96%  <256>    [72.0, 44.0]
  // BM_map_string_delete       3168     2080  +34.34%  <256>    [72.0, 55.8]
  // BM_map_string_queueaddrem  5840     4701  +19.50%  <256>    [72.0, 59.4]
  // BM_map_string_mixedaddrem  6400     5200  +18.75%  <256>    [72.0, 57.8]
  // BM_map_string_fifo          398      596  -49.75%  <256>    [72.0, 44.0]
  // BM_map_string_fwditer       243      113  +53.50%  <256>    [72.0, 55.8]
  
  #ifndef UTIL_BTREE_BTREE_H__
  #define UTIL_BTREE_BTREE_H__
  
  #include <assert.h>
  #include <stddef.h>
  #include <string.h>
  #include <sys/types.h>
  #include <algorithm>
  #include <functional>
  #include <iostream>
  #include <iterator>
  #include <limits>
  #include <type_traits>
  #include <new>
  #include <ostream>
  #include <string>
  #include <utility>
  
  #ifndef NDEBUG
  #define NDEBUG 1
  #endif
  
  namespace btree {
  
  // Inside a btree method, if we just call swap(), it will choose the
  // btree::swap method, which we don't want. And we can't say ::swap
  // because then MSVC won't pickup any std::swap() implementations. We
  // can't just use std::swap() directly because then we don't get the
  // specialization for types outside the std namespace. So the solution
  // is to have a special swap helper function whose name doesn't
  // collide with other swap functions defined by the btree classes.
  template <typename T>
  inline void btree_swap_helper(T &a, T &b) {
    using std::swap;
    swap(a, b);
  }
  
  // A template helper used to select A or B based on a condition.
  template<bool cond, typename A, typename B>
  struct if_{
    typedef A type;
  };
  
  template<typename A, typename B>
  struct if_<false, A, B> {
    typedef B type;
  };
  
  // Types small_ and big_ are promise that sizeof(small_) < sizeof(big_)
  typedef char small_;
  
  struct big_ {
    char dummy[2];
  };
  
  // A compile-time assertion.
  template <bool>
  struct CompileAssert {
  };
  
  #define COMPILE_ASSERT(expr, msg) \
    typedef CompileAssert<(bool(expr))> msg[bool(expr) ? 1 : -1]
  
  // A helper type used to indicate that a key-compare-to functor has been
  // provided. A user can specify a key-compare-to functor by doing:
  //
  //  struct MyStringComparer
  //      : public util::btree::btree_key_compare_to_tag {
  //    int operator()(const string &a, const string &b) const {
  //      return a.compare(b);
  //    }
  //  };
  //
  // Note that the return type is an int and not a bool. There is a
  // COMPILE_ASSERT which enforces this return type.
  struct btree_key_compare_to_tag {
  };
  
  // A helper class that indicates if the Compare parameter is derived from
  // btree_key_compare_to_tag.
  template <typename Compare>
  struct btree_is_key_compare_to
      : public std::is_convertible<Compare, btree_key_compare_to_tag> {
  };
  
  // 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<string> and greater<string>. The btree_key_compare_to_adapter
  // class is provided so that btree users automatically get the more
  // efficient compare-to code when using common google string types
  // with common comparison functors.
  template <typename Compare>
  struct btree_key_compare_to_adapter : Compare {
    btree_key_compare_to_adapter() { }
    btree_key_compare_to_adapter(const Compare &c) : Compare(c) { }
    btree_key_compare_to_adapter(const btree_key_compare_to_adapter<Compare> &c)
        : Compare(c) {
    }
  };
  
  template <>
  struct btree_key_compare_to_adapter<std::less<std::string> >
      : public btree_key_compare_to_tag {
    btree_key_compare_to_adapter() {}
    btree_key_compare_to_adapter(const std::less<std::string>&) {}
    btree_key_compare_to_adapter(
        const btree_key_compare_to_adapter<std::less<std::string> >&) {}
    int operator()(const std::string &a, const std::string &b) const {
      return a.compare(b);
    }
  };
  
  template <>
  struct btree_key_compare_to_adapter<std::greater<std::string> >
      : public btree_key_compare_to_tag {
    btree_key_compare_to_adapter() {}
    btree_key_compare_to_adapter(const std::greater<std::string>&) {}
    btree_key_compare_to_adapter(
        const btree_key_compare_to_adapter<std::greater<std::string> >&) {}
    int operator()(const std::string &a, const std::string &b) const {
      return b.compare(a);
    }
  };
  
  // A helper class that allows a compare-to functor to behave like a plain
  // compare functor. This specialization is used when we do not have a
  // compare-to functor.
  template <typename Key, typename Compare, bool HaveCompareTo>
  struct btree_key_comparer {
    btree_key_comparer() {}
    btree_key_comparer(Compare c) : comp(c) {}
    static bool bool_compare(const Compare &comp, const Key &x, const Key &y) {
      return comp(x, y);
    }
    bool operator()(const Key &x, const Key &y) const {
      return bool_compare(comp, x, y);
    }
    Compare comp;
  };
  
  // A specialization of btree_key_comparer when a compare-to functor is
  // present. We need a plain (boolean) comparison in some parts of the btree
  // code, such as insert-with-hint.
  template <typename Key, typename Compare>
  struct btree_key_comparer<Key, Compare, true> {
    btree_key_comparer() {}
    btree_key_comparer(Compare c) : comp(c) {}
    static bool bool_compare(const Compare &comp, const Key &x, const Key &y) {
      return comp(x, y) < 0;
    }
    bool operator()(const Key &x, const Key &y) const {
      return bool_compare(comp, x, y);
    }
    Compare comp;
  };
  
  // A helper function to compare to keys using the specified compare
  // functor. This dispatches to the appropriate btree_key_comparer comparison,
  // depending on whether we have a compare-to functor or not (which depends on
  // whether Compare is derived from btree_key_compare_to_tag).
  template <typename Key, typename Compare>
  static bool btree_compare_keys(
      const Compare &comp, const Key &x, const Key &y) {
    typedef btree_key_comparer<Key, Compare,
        btree_is_key_compare_to<Compare>::value> key_comparer;
    return key_comparer::bool_compare(comp, x, y);
  }
  
  template <typename Key, typename Compare,
            typename Alloc, int TargetNodeSize, int ValueSize>
  struct btree_common_params {
    // If Compare is derived from btree_key_compare_to_tag then use it as the
    // key_compare type. Otherwise, use btree_key_compare_to_adapter<> which will
    // fall-back to Compare if we don't have an appropriate specialization.
    typedef typename if_<
      btree_is_key_compare_to<Compare>::value,
      Compare, btree_key_compare_to_adapter<Compare> >::type key_compare;
    // A type which indicates if we have a key-compare-to functor or a plain old
    // key-compare functor.
    typedef btree_is_key_compare_to<key_compare> is_key_compare_to;
  
    typedef Alloc allocator_type;
    typedef Key key_type;
    typedef ssize_t size_type;
    typedef ptrdiff_t difference_type;
  
    enum {
      kTargetNodeSize = TargetNodeSize,
  
      // Available space for values.  This is largest for leaf nodes,
      // which has overhead no fewer than two pointers.
      kNodeValueSpace = TargetNodeSize - 2 * sizeof(void*),
    };
  
    // This is an integral type large enough to hold as many
    // ValueSize-values as will fit a node of TargetNodeSize bytes.
    typedef typename if_<
      (kNodeValueSpace / ValueSize) >= 256,
      uint16_t,
      uint8_t>::type node_count_type;
  };
  
  // A parameters structure for holding the type parameters for a btree_map.
  template <typename Key, typename Data, typename Compare,
            typename Alloc, int TargetNodeSize>
  struct btree_map_params
      : public btree_common_params<Key, Compare, Alloc, TargetNodeSize,
                                   sizeof(Key) + sizeof(Data)> {
    typedef Data data_type;
    typedef Data mapped_type;
    typedef std::pair<const Key, data_type> value_type;
    typedef std::pair<Key, data_type> mutable_value_type;
    typedef value_type* pointer;
    typedef const value_type* const_pointer;
    typedef value_type& reference;
    typedef const value_type& const_reference;
  
    enum {
      kValueSize = sizeof(Key) + sizeof(data_type),
    };
  
    static const Key& key(const value_type &x) { return x.first; }
    static const Key& key(const mutable_value_type &x) { return x.first; }
    static void swap(mutable_value_type *a, mutable_value_type *b) {
      btree_swap_helper(a->first, b->first);
      btree_swap_helper(a->second, b->second);
    }
  };
  
  // A parameters structure for holding the type parameters for a btree_set.
  template <typename Key, typename Compare, typename Alloc, int TargetNodeSize>
  struct btree_set_params
      : public btree_common_params<Key, Compare, Alloc, TargetNodeSize,
                                   sizeof(Key)> {
    typedef std::false_type data_type;
    typedef std::false_type mapped_type;
    typedef Key value_type;
    typedef value_type mutable_value_type;
    typedef value_type* pointer;
    typedef const value_type* const_pointer;
    typedef value_type& reference;
    typedef const value_type& const_reference;
  
    enum {
      kValueSize = sizeof(Key),
    };
  
    static const Key& key(const value_type &x) { return x; }
    static void swap(mutable_value_type *a, mutable_value_type *b) {
      btree_swap_helper<mutable_value_type>(*a, *b);
    }
  };
  
  // An adapter class that converts a lower-bound compare into an upper-bound
  // compare.
  template <typename Key, typename Compare>
  struct btree_upper_bound_adapter : public Compare {
    btree_upper_bound_adapter(Compare c) : Compare(c) {}
    bool operator()(const Key &a, const Key &b) const {
      return !static_cast<const Compare&>(*this)(b, a);
    }
  };
  
  template <typename Key, typename CompareTo>
  struct btree_upper_bound_compare_to_adapter : public CompareTo {
    btree_upper_bound_compare_to_adapter(CompareTo c) : CompareTo(c) {}
    int operator()(const Key &a, const Key &b) const {
      return static_cast<const CompareTo&>(*this)(b, a);
    }
  };
  
  // Dispatch helper class for using linear search with plain compare.
  template <typename K, typename N, typename Compare>
  struct btree_linear_search_plain_compare {
    static int lower_bound(const K &k, const N &n, Compare comp)  {
      return n.linear_search_plain_compare(k, 0, n.count(), comp);
    }
    static int upper_bound(const K &k, const N &n, Compare comp)  {
      typedef btree_upper_bound_adapter<K, Compare> upper_compare;
      return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
    }
  };
  
  // Dispatch helper class for using linear search with compare-to
  template <typename K, typename N, typename CompareTo>
  struct btree_linear_search_compare_to {
    static int lower_bound(const K &k, const N &n, CompareTo comp)  {
      return n.linear_search_compare_to(k, 0, n.count(), comp);
    }
    static int upper_bound(const K &k, const N &n, CompareTo comp)  {
      typedef btree_upper_bound_adapter<K,
          btree_key_comparer<K, CompareTo, true> > upper_compare;
      return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
    }
  };
  
  // Dispatch helper class for using binary search with plain compare.
  template <typename K, typename N, typename Compare>
  struct btree_binary_search_plain_compare {
    static int lower_bound(const K &k, const N &n, Compare comp)  {
      return n.binary_search_plain_compare(k, 0, n.count(), comp);
    }
    static int upper_bound(const K &k, const N &n, Compare comp)  {
      typedef btree_upper_bound_adapter<K, Compare> upper_compare;
      return n.binary_search_plain_compare(k, 0, n.count(), upper_compare(comp));
    }
  };
  
  // Dispatch helper class for using binary search with compare-to.
  template <typename K, typename N, typename CompareTo>
  struct btree_binary_search_compare_to {
    static int lower_bound(const K &k, const N &n, CompareTo comp)  {
      return n.binary_search_compare_to(k, 0, n.count(), CompareTo());
    }
    static int upper_bound(const K &k, const N &n, CompareTo comp)  {
      typedef btree_upper_bound_adapter<K,
          btree_key_comparer<K, CompareTo, true> > upper_compare;
      return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
    }
  };
  
  // 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 {
   public:
    typedef Params params_type;
    typedef btree_node<Params> self_type;
    typedef typename Params::key_type key_type;
    typedef typename Params::data_type data_type;
    typedef typename Params::value_type value_type;
    typedef typename Params::mutable_value_type mutable_value_type;
    typedef typename Params::pointer pointer;
    typedef typename Params::const_pointer const_pointer;
    typedef typename Params::reference reference;
    typedef typename Params::const_reference const_reference;
    typedef typename Params::key_compare key_compare;
    typedef typename Params::size_type size_type;
    typedef typename Params::difference_type difference_type;
    // Typedefs for the various types of node searches.
    typedef btree_linear_search_plain_compare<
      key_type, self_type, key_compare> linear_search_plain_compare_type;
    typedef btree_linear_search_compare_to<
      key_type, self_type, key_compare> linear_search_compare_to_type;
    typedef btree_binary_search_plain_compare<
      key_type, self_type, key_compare> binary_search_plain_compare_type;
    typedef btree_binary_search_compare_to<
      key_type, self_type, key_compare> binary_search_compare_to_type;
    // If we have a valid key-compare-to type, use linear_search_compare_to,
    // otherwise use linear_search_plain_compare.
    typedef typename if_<
      Params::is_key_compare_to::value,
      linear_search_compare_to_type,
      linear_search_plain_compare_type>::type linear_search_type;
    // If we have a valid key-compare-to type, use binary_search_compare_to,
    // otherwise use binary_search_plain_compare.
    typedef typename if_<
      Params::is_key_compare_to::value,
      binary_search_compare_to_type,
      binary_search_plain_compare_type>::type binary_search_type;
    // If the key is an integral or floating point type, use linear search which
    // is faster than binary search for such types. Might be wise to also
    // configure linear search based on node-size.
    typedef typename if_<
      std::is_integral<key_type>::value ||
      std::is_floating_point<key_type>::value,
      linear_search_type, binary_search_type>::type search_type;
  
    struct base_fields {
      typedef typename Params::node_count_type field_type;
  
      // A boolean indicating whether the node is a leaf or not.
      bool leaf;
      // The position of the node in the node's parent.
      field_type position;
      // The maximum number of values the node can hold.
      field_type max_count;
      // The count of the number of values in the node.
      field_type count;
      // A pointer to the node's parent.
      btree_node *parent;
    };
  
    enum {
      kValueSize = params_type::kValueSize,
      kTargetNodeSize = params_type::kTargetNodeSize,
  
      // Compute how many values we can fit onto a leaf node.
      kNodeTargetValues = (kTargetNodeSize - sizeof(base_fields)) / kValueSize,
      // 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,
  
      kExactMatch = 1 << 30,
      kMatchMask = kExactMatch - 1,
    };
  
    struct leaf_fields : public base_fields {
      // The array of values. Only the first count of these values have been
      // constructed and are valid.
      mutable_value_type values[kNodeValues];
    };
  
    struct internal_fields : public leaf_fields {
      // 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 always count + 1 children.
      btree_node *children[kNodeValues + 1];
    };
  
    struct root_fields : public internal_fields {
      btree_node *rightmost;
      size_type size;
    };
  
   public:
    // Getter/setter for whether this is a leaf node or not. This value doesn't
    // change after the node is created.
    bool leaf() const { return fields_.leaf; }
  
    // Getter for the position of this node in its parent.
    int position() const { return fields_.position; }
    void set_position(int v) { fields_.position = v; }
  
    // Getter/setter for the number of values stored in this node.
    int count() const { return fields_.count; }
    void set_count(int v) { fields_.count = v; }
    int max_count() const { return fields_.max_count; }
  
    // Getter for the parent of this node.
    btree_node* parent() const { return fields_.parent; }
    // 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());
      fields_.parent = fields_.parent->parent();
    }
  
    // Getter for the rightmost root node field. Only valid on the root node.
    btree_node* rightmost() const { return fields_.rightmost; }
    btree_node** mutable_rightmost() { return &fields_.rightmost; }
  
    // Getter for the size root node field. Only valid on the root node.
    size_type size() const { return fields_.size; }
    size_type* mutable_size() { return &fields_.size; }
  
    // Getters for the key/value at position i in the node.
    const key_type& key(int i) const {
      return params_type::key(fields_.values[i]);
    }
    reference value(int i) {
      return reinterpret_cast<reference>(fields_.values[i]);
    }
    const_reference value(int i) const {
      return reinterpret_cast<const_reference>(fields_.values[i]);
    }
    mutable_value_type* mutable_value(int i) {
      return &fields_.values[i];
    }
  
    // Swap value i in this node with value j in node x.
    void value_swap(int i, btree_node *x, int j) {
      params_type::swap(mutable_value(i), x->mutable_value(j));
    }
  
    // Getters/setter for the child at position i in the node.
    btree_node* child(int i) const { return fields_.children[i]; }
    btree_node** mutable_child(int i) { return &fields_.children[i]; }
    void set_child(int i, btree_node *c) {
      *mutable_child(i) = c;
      c->fields_.parent = this;
      c->fields_.position = i;
    }
  
    // Returns the position of the first value whose key is not less than k.
    template <typename Compare>
    int lower_bound(const key_type &k, const Compare &comp) const {
      return search_type::lower_bound(k, *this, comp);
    }
    // Returns the position of the first value whose key is greater than k.
    template <typename Compare>
    int upper_bound(const key_type &k, const Compare &comp) const {
      return search_type::upper_bound(k, *this, comp);
    }
  
    // Returns the position of the first value whose key is not less than k using
    // linear search performed using plain compare.
    template <typename Compare>
    int linear_search_plain_compare(
        const key_type &k, int s, int e, const Compare &comp) const {
      while (s < e) {
        if (!btree_compare_keys(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 Compare>
    int linear_search_compare_to(
        const key_type &k, int s, int e, const Compare &comp) const {
      while (s < e) {
        int c = comp(key(s), k);
        if (c == 0) {
          return s | kExactMatch;
        } else if (c > 0) {
          break;
        }
        ++s;
      }
      return s;
    }
  
    // Returns the position of the first value whose key is not less than k using
    // binary search performed using plain compare.
    template <typename Compare>
    int binary_search_plain_compare(
        const key_type &k, int s, int e, const Compare &comp) const {
      while (s != e) {
        int mid = (s + e) / 2;
        if (btree_compare_keys(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 CompareTo>
    int binary_search_compare_to(
        const key_type &k, int s, int e, const CompareTo &comp) const {
      while (s != e) {
        int mid = (s + e) / 2;
        int c = comp(key(mid), k);
        if (c < 0) {
          s = mid + 1;
        } else if (c > 0) {
          e = mid;
        } else {
          // Need to return the first value whose key is not less than k, which
          // requires continuing the binary search. Note that we are guaranteed
          // that the result is an exact match because if "key(mid-1) < k" the
          // call to binary_search_compare_to() will return "mid".
          s = binary_search_compare_to(k, s, mid, comp);
          return s | kExactMatch;
        }
      }
      return s;
    }
  
    // Inserts the value x at position i, shifting all existing values and
    // children at positions >= i to the right by 1.
    void insert_value(int i, const value_type &x);
  
    // 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);
  
    // Rebalances a node with its right sibling.
    void rebalance_right_to_left(btree_node *sibling, int to_move);
    void rebalance_left_to_right(btree_node *sibling, int to_move);
  
    // Splits a node, moving a portion of the node's values to its right sibling.
    void split(btree_node *sibling, int insert_position);
  
    // 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 *sibling);
  
    // Swap the contents of "this" and "src".
    void swap(btree_node *src);
  
    // Node allocation/deletion routines.
    static btree_node* init_leaf(
        leaf_fields *f, btree_node *parent, int max_count) {
      btree_node *n = reinterpret_cast<btree_node*>(f);
      f->leaf = 1;
      f->position = 0;
      f->max_count = max_count;
      f->count = 0;
      f->parent = parent;
      if (!NDEBUG) {
        memset(&f->values, 0, max_count * sizeof(value_type));
      }
      return n;
    }
    static btree_node* init_internal(internal_fields *f, btree_node *parent) {
      btree_node *n = init_leaf(f, parent, kNodeValues);
      f->leaf = 0;
      if (!NDEBUG) {
        memset(f->children, 0, sizeof(f->children));
      }
      return n;
    }
    static btree_node* init_root(root_fields *f, btree_node *parent) {
      btree_node *n = init_internal(f, parent);
      f->rightmost = parent;
      f->size = parent->count();
      return n;
    }
    void destroy() {
      for (int i = 0; i < count(); ++i) {
        value_destroy(i);
      }
    }
  
   private:
    void value_init(int i) {
      new (&fields_.values[i]) mutable_value_type;
    }
    void value_init(int i, const value_type &x) {
      new (&fields_.values[i]) mutable_value_type(x);
    }
    void value_destroy(int i) {
      fields_.values[i].~mutable_value_type();
    }
  
   private:
    root_fields fields_;
  
   private:
    btree_node(const btree_node&);
    void operator=(const btree_node&);
  };
  
  template <typename Node, typename Reference, typename Pointer>
  struct btree_iterator {
    typedef typename Node::key_type key_type;
    typedef typename Node::size_type size_type;
    typedef typename Node::difference_type difference_type;
    typedef typename Node::params_type params_type;
  
    typedef Node node_type;
    typedef typename std::remove_const<Node>::type normal_node;
    typedef const Node const_node;
    typedef typename params_type::value_type value_type;
    typedef typename params_type::pointer normal_pointer;
    typedef typename params_type::reference normal_reference;
    typedef typename params_type::const_pointer const_pointer;
    typedef typename params_type::const_reference const_reference;
  
    typedef Pointer pointer;
    typedef Reference reference;
    typedef std::bidirectional_iterator_tag iterator_category;
  
    typedef btree_iterator<
      normal_node, normal_reference, normal_pointer> iterator;
    typedef btree_iterator<
      const_node, const_reference, const_pointer> const_iterator;
    typedef btree_iterator<Node, Reference, Pointer> self_type;
  
    btree_iterator()
        : node(NULL),
          position(-1) {
    }
    btree_iterator(Node *n, int p)
        : node(n),
          position(p) {
    }
    btree_iterator(const iterator &x)
        : node(x.node),
          position(x.position) {
    }
  
    // Increment/decrement the iterator.
    void increment() {
      if (node->leaf() && ++position < node->count()) {
        return;
      }
      increment_slow();
    }
    void increment_by(int count);
    void increment_slow();
  
    void decrement() {
      if (node->leaf() && --position >= 0) {
        return;
      }
      decrement_slow();
    }
    void decrement_slow();
  
    bool operator==(const const_iterator &x) const {
      return node == x.node && position == x.position;
    }
    bool operator!=(const const_iterator &x) const {
      return node != x.node || position != x.position;
    }
  
    // Accessors for the key/value the iterator is pointing at.
    const key_type& key() const {
      return node->key(position);
    }
    reference operator*() const {
      return node->value(position);
    }
    pointer operator->() const {
      return &node->value(position);
    }
  
    self_type& operator++() {
      increment();
      return *this;
    }
    self_type& operator--() {
      decrement();
      return *this;
    }
    self_type operator++(int) {
      self_type tmp = *this;
      ++*this;
      return tmp;
    }
    self_type operator--(int) {
      self_type tmp = *this;
      --*this;
      return tmp;
    }
  
    // 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.
    int position;
  };
  
  // Dispatch helper class for using btree::internal_locate with plain compare.
  struct btree_internal_locate_plain_compare {
    template <typename K, typename T, typename Iter>
    static std::pair<Iter, int> dispatch(const K &k, const T &t, Iter iter) {
      return t.internal_locate_plain_compare(k, iter);
    }
  };
  
  // Dispatch helper class for using btree::internal_locate with compare-to.
  struct btree_internal_locate_compare_to {
    template <typename K, typename T, typename Iter>
    static std::pair<Iter, int> dispatch(const K &k, const T &t, Iter iter) {
      return t.internal_locate_compare_to(k, iter);
    }
  };
  
  template <typename Params>
  class btree : public Params::key_compare {
    typedef btree<Params> self_type;
    typedef btree_node<Params> node_type;
    typedef typename node_type::base_fields base_fields;
    typedef typename node_type::leaf_fields leaf_fields;
    typedef typename node_type::internal_fields internal_fields;
    typedef typename node_type::root_fields root_fields;
    typedef typename Params::is_key_compare_to is_key_compare_to;
  
    friend struct btree_internal_locate_plain_compare;
    friend struct btree_internal_locate_compare_to;
    typedef typename if_<
      is_key_compare_to::value,
      btree_internal_locate_compare_to,
      btree_internal_locate_plain_compare>::type internal_locate_type;
  
    enum {
      kNodeValues = node_type::kNodeValues,
      kMinNodeValues = kNodeValues / 2,
      kValueSize = node_type::kValueSize,
      kExactMatch = node_type::kExactMatch,
      kMatchMask = node_type::kMatchMask,
    };
  
    // A helper class to get the empty base class optimization for 0-size
    // allocators. Base is internal_allocator_type.
    // (e.g. empty_base_handle<internal_allocator_type, node_type*>). If Base is
    // 0-size, the compiler doesn't have to reserve any space for it and
    // sizeof(empty_base_handle) will simply be sizeof(Data). Google [empty base
    // class optimization] for more details.
    template <typename Base, typename Data>
    struct empty_base_handle : public Base {
      empty_base_handle(const Base &b, const Data &d)
          : Base(b),
            data(d) {
      }
      Data data;
    };
  
    struct node_stats {
      node_stats(ssize_t l, ssize_t i)
          : leaf_nodes(l),
            internal_nodes(i) {
      }
  
      node_stats& operator+=(const node_stats &x) {
        leaf_nodes += x.leaf_nodes;
        internal_nodes += x.internal_nodes;
        return *this;
      }
  
      ssize_t leaf_nodes;
      ssize_t internal_nodes;
    };
  
   public:
    typedef Params params_type;
    typedef typename Params::key_type key_type;
    typedef typename Params::data_type data_type;
    typedef typename Params::mapped_type mapped_type;
    typedef typename Params::value_type value_type;
    typedef typename Params::key_compare key_compare;
    typedef typename Params::pointer pointer;
    typedef typename Params::const_pointer const_pointer;
    typedef typename Params::reference reference;
    typedef typename Params::const_reference const_reference;
    typedef typename Params::size_type size_type;
    typedef typename Params::difference_type difference_type;
    typedef btree_iterator<node_type, reference, pointer> iterator;
    typedef typename iterator::const_iterator const_iterator;
    typedef std::reverse_iterator<const_iterator> const_reverse_iterator;
    typedef std::reverse_iterator<iterator> reverse_iterator;
  
    typedef typename Params::allocator_type allocator_type;
    typedef typename allocator_type::template rebind<char>::other
      internal_allocator_type;
  
   public:
    // Default constructor.
    btree(const key_compare &comp, const allocator_type &alloc);
  
    // Copy constructor.
    btree(const self_type &x);
  
    // Destructor.
    ~btree() {
      clear();
    }
  
    // Iterator routines.
    iterator begin() {
      return iterator(leftmost(), 0);
    }
    const_iterator begin() const {
      return const_iterator(leftmost(), 0);
    }
    iterator end() {
      return iterator(rightmost(), rightmost() ? rightmost()->count() : 0);
    }
    const_iterator end() const {
      return const_iterator(rightmost(), rightmost() ? rightmost()->count() : 0);
    }
    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.
    iterator lower_bound(const key_type &key) {
      return internal_end(
          internal_lower_bound(key, iterator(root(), 0)));
    }
    const_iterator lower_bound(const key_type &key) const {
      return internal_end(
          internal_lower_bound(key, const_iterator(root(), 0)));
    }
  
    // Finds the first element whose key is greater than key.
    iterator upper_bound(const key_type &key) {
      return internal_end(
          internal_upper_bound(key, iterator(root(), 0)));
    }
    const_iterator upper_bound(const key_type &key) const {
      return internal_end(
          internal_upper_bound(key, const_iterator(root(), 0)));
    }
  
    // 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).
    std::pair<iterator,iterator> equal_range(const key_type &key) {
      return std::make_pair(lower_bound(key), upper_bound(key));
    }
    std::pair<const_iterator,const_iterator> equal_range(const key_type &key) const {
      return std::make_pair(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. The
    // ValuePointer type is used to avoid instatiating the value unless the key
    // is being inserted. Value is not dereferenced if the key already exists in
    // the btree. See btree_map::operator[].
    template <typename ValuePointer>
    std::pair<iterator,bool> insert_unique(const key_type &key, ValuePointer value);
  
    // Inserts a value into the btree only if it does not already exist. The
    // boolean return value indicates whether insertion succeeded or failed.
    std::pair<iterator,bool> insert_unique(const value_type &v) {
      return insert_unique(params_type::key(v), &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_unique(v) were made.
    iterator insert_unique(iterator position, const value_type &v);
  
    // Insert a range of values into the btree.
    template <typename InputIterator>
    void insert_unique(InputIterator b, InputIterator e);
  
    // Inserts a value into the btree. The ValuePointer type is used to avoid
    // instatiating the value unless the key is being inserted. Value is not
    // dereferenced if the key already exists in the btree. See
    // btree_map::operator[].
    template <typename ValuePointer>
    iterator insert_multi(const key_type &key, ValuePointer value);
  
    // Inserts a value into the btree.
    iterator insert_multi(const value_type &v) {
      return insert_multi(params_type::key(v), &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.
    iterator insert_multi(iterator position, const value_type &v);
  
    // Insert a range of values into the btree.
    template <typename InputIterator>
    void insert_multi(InputIterator b, InputIterator e);
  
    void assign(const self_type &x);
  
    // 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).
    iterator erase(iterator iter);
  
    // Erases range. Returns the number of keys erased.
    int erase(iterator begin, iterator end);
  
    // Erases the specified key from the btree. Returns 1 if an element was
    // erased and 0 otherwise.
    int erase_unique(const key_type &key);
  
    // Erases all of the entries matching the specified key from the
    // btree. Returns the number of elements erased.
    int erase_multi(const key_type &key);
  
    // Finds the iterator corresponding to a key or returns end() if the key is
    // not present.
    iterator find_unique(const key_type &key) {
      return internal_end(
          internal_find_unique(key, iterator(root(), 0)));
    }
    const_iterator find_unique(const key_type &key) const {
      return internal_end(
          internal_find_unique(key, const_iterator(root(), 0)));
    }
    iterator find_multi(const key_type &key) {
      return internal_end(
          internal_find_multi(key, iterator(root(), 0)));
    }
    const_iterator find_multi(const key_type &key) const {
      return internal_end(
          internal_find_multi(key, const_iterator(root(), 0)));
    }
  
    // Returns a count of the number of times the key appears in the btree.
    size_type count_unique(const key_type &key) const {
      const_iterator b = internal_find_unique(
          key, const_iterator(root(), 0));
      if (!b.node) {
        // 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.
    size_type count_multi(const key_type &key) const {
      return distance(lower_bound(key), upper_bound(key));
    }
  
    // Clear the btree, deleting all of the values it contains.
    void clear();
  
    // Swap the contents of *this and x.
    void swap(self_type &x);
  
    // Assign the contents of x to *this.
    self_type& operator=(const self_type &x) {
      if (&x == this) {
        // Don't copy onto ourselves.
        return *this;
      }
      assign(x);
      return *this;
    }
  
    key_compare* mutable_key_comp() {
      return this;
    }
    const key_compare& key_comp() const {
      return *this;
    }
    bool compare_keys(const key_type &x, const key_type &y) const {
      return btree_compare_keys(key_comp(), x, y);
    }
  
    // Dump the btree to the specified ostream. Requires that operator<< is
    // defined for Key and Value.
    void dump(std::ostream &os) const {
      if (root() != NULL) {
        internal_dump(os, root(), 0);
      }
    }
  
    // Verifies the structure of the btree.
    void verify() const;
  
    // Size routines. Note that empty() is slightly faster than doing size()==0.
    size_type size() const {
      if (empty()) return 0;
      if (root()->leaf()) return root()->count();
      return root()->size();
    }
    size_type max_size() const { return std::numeric_limits<size_type>::max(); }
    bool empty() const { return root() == NULL; }
  
    // The height of the btree. An empty tree will have height 0.
    size_type height() const {
      size_type h = 0;
      if (root()) {
        // 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) +
            sizeof(base_fields) + root()->max_count() * sizeof(value_type);
      } else {
        return sizeof(*this) +
            sizeof(root_fields) - sizeof(internal_fields) +
            stats.leaf_nodes * sizeof(leaf_fields) +
            stats.internal_nodes * sizeof(internal_fields);
      }
    }
  
    // 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 sizeof(leaf_fields) / (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.
    double fullness() const {
      return 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.
    double overhead() const {
      if (empty()) {
        return 0.0;
      }
      return (bytes_used() - size() * kValueSize) / double(size());
    }
  
   private:
    // Internal accessor routines.
    node_type* root() { return root_.data; }
    const node_type* root() const { return root_.data; }
    node_type** mutable_root() { return &root_.data; }
  
    // The rightmost node is stored in the root node.
    node_type* rightmost() {
      return (!root() || root()->leaf()) ? root() : root()->rightmost();
    }
    const node_type* rightmost() const {
      return (!root() || root()->leaf()) ? root() : root()->rightmost();
    }
    node_type** mutable_rightmost() { return root()->mutable_rightmost(); }
  
    // The leftmost node is stored as the parent of the root node.
    node_type* leftmost() { return root() ? root()->parent() : NULL; }
    const node_type* leftmost() const { return root() ? root()->parent() : NULL; }
  
    // The size of the tree is stored in the root node.
    size_type* mutable_size() { return root()->mutable_size(); }
  
    // Allocator routines.
    internal_allocator_type* mutable_internal_allocator() {
      return static_cast<internal_allocator_type*>(&root_);
    }
    const internal_allocator_type& internal_allocator() const {
      return *static_cast<const internal_allocator_type*>(&root_);
    }
  
    // Node creation/deletion routines.
    node_type* new_internal_node(node_type *parent) {
      internal_fields *p = reinterpret_cast<internal_fields*>(
          mutable_internal_allocator()->allocate(sizeof(internal_fields)));
      return node_type::init_internal(p, parent);
    }
    node_type* new_internal_root_node() {
      root_fields *p = reinterpret_cast<root_fields*>(
          mutable_internal_allocator()->allocate(sizeof(root_fields)));
      return node_type::init_root(p, root()->parent());
    }
    node_type* new_leaf_node(node_type *parent) {
      leaf_fields *p = reinterpret_cast<leaf_fields*>(
          mutable_internal_allocator()->allocate(sizeof(leaf_fields)));
      return node_type::init_leaf(p, parent, kNodeValues);
    }
    node_type* new_leaf_root_node(int max_count) {
      leaf_fields *p = reinterpret_cast<leaf_fields*>(
          mutable_internal_allocator()->allocate(
              sizeof(base_fields) + max_count * sizeof(value_type)));
      return node_type::init_leaf(p, reinterpret_cast<node_type*>(p), max_count);
    }
    void delete_internal_node(node_type *node) {
      node->destroy();
      assert(node != root());
      mutable_internal_allocator()->deallocate(
          reinterpret_cast<char*>(node), sizeof(internal_fields));
    }
    void delete_internal_root_node() {
      root()->destroy();
      mutable_internal_allocator()->deallocate(
          reinterpret_cast<char*>(root()), sizeof(root_fields));
    }
    void delete_leaf_node(node_type *node) {
      node->destroy();
      mutable_internal_allocator()->deallocate(
          reinterpret_cast<char*>(node),
          sizeof(base_fields) + node->max_count() * sizeof(value_type));
    }
  
    // 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 ? iter : end();
    }
    const_iterator internal_end(const_iterator iter) const {
      return iter.node ? iter : end();
    }
  
    // Inserts a value into the btree immediately before iter. Requires that
    // key(v) <= iter.key() and (--iter).key() <= key(v).
    iterator internal_insert(iterator iter, const value_type &v);
  
    // Returns an iterator pointing to the first value >= the value "iter" is
    // pointing at. Note that "iter" might be pointing to an invalid location as
    // iter.position == iter.node->count(). This routine simply moves iter up in
    // the tree to a valid location.
    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 (internal_locate_plain_compare) always returns 0 for the second
    // field of the pair. The second version (internal_locate_compare_to) is for
    // the key-compare-to specialization and returns either kExactMatch (if the
    // key was found in the tree) or -kExactMatch (if it wasn't) in the second
    // field of the pair. The compare_to specialization allows the caller to
    // avoid a subsequent comparison to determine if an exact match was made,
    // speeding up string keys.
    template <typename IterType>
    std::pair<IterType, int> internal_locate(
        const key_type &key, IterType iter) const;
    template <typename IterType>
    std::pair<IterType, int> internal_locate_plain_compare(
        const key_type &key, IterType iter) const;
    template <typename IterType>
    std::pair<IterType, int> internal_locate_compare_to(
        const key_type &key, IterType iter) const;
  
    // Internal routine which implements lower_bound().
    template <typename IterType>
    IterType internal_lower_bound(
        const key_type &key, IterType iter) const;
  
    // Internal routine which implements upper_bound().
    template <typename IterType>
    IterType internal_upper_bound(
        const key_type &key, IterType iter) const;
  
    // Internal routine which implements find_unique().
    template <typename IterType>
    IterType internal_find_unique(
        const key_type &key, IterType iter) const;
  
    // Internal routine which implements find_multi().
    template <typename IterType>
    IterType internal_find_multi(
        const key_type &key, IterType iter) const;
  
    // Deletes a node and all of its children.
    void internal_clear(node_type *node);
  
    // Dumps a node and all of its children to the specified ostream.
    void internal_dump(std::ostream &os, const node_type *node, int level) const;
  
    // 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 {
      if (!node) {
        return node_stats(0, 0);
      }
      if (node->leaf()) {
        return node_stats(1, 0);
      }
      node_stats res(0, 1);
      for (int i = 0; i <= node->count(); ++i) {
        res += internal_stats(node->child(i));
      }
      return res;
    }
  
   private:
    empty_base_handle<internal_allocator_type, node_type*> root_;
  
   private:
    // A never instantiated helper function that returns big_ if we have a
    // key-compare-to functor or if R is bool and small_ otherwise.
    template <typename R>
    static typename if_<
     if_<is_key_compare_to::value,
               std::is_same<R, int>,
               std::is_same<R, bool> >::type::value,
     big_, small_>::type key_compare_checker(R);
  
    // A never instantiated helper function that returns the key comparison
    // functor.
    static key_compare key_compare_helper();
  
    // Verify that key_compare returns a bool. This is similar to the way
    // is_convertible in base/type_traits.h works. Note that key_compare_checker
    // is never actually invoked. The compiler will select which
    // key_compare_checker() to instantiate and then figure out the size of the
    // return type of key_compare_checker() at compile time which we then check
    // against the sizeof of big_.
    COMPILE_ASSERT(
        sizeof(key_compare_checker(key_compare_helper()(key_type(), key_type()))) ==
        sizeof(big_),
        key_comparison_function_must_return_bool);
  
    // Note: We insist on kTargetValues, which is computed from
    // Params::kTargetNodeSize, must fit the base_fields::field_type.
    COMPILE_ASSERT(kNodeValues <
                   (1 << (8 * sizeof(typename base_fields::field_type))),
                   target_node_size_too_large);
  
    // Test the assumption made in setting kNodeValueSpace.
    COMPILE_ASSERT(sizeof(base_fields) >= 2 * sizeof(void*),
                   node_space_assumption_incorrect);
  };
  
  ////
  // btree_node methods
  template <typename P>
  inline void btree_node<P>::insert_value(int i, const value_type &x) {
    assert(i <= count());
    value_init(count(), x);
    for (int j = count(); j > i; --j) {
      value_swap(j, this, j - 1);
    }
    set_count(count() + 1);
  
    if (!leaf()) {
      ++i;
      for (int j = count(); j > i; --j) {
        *mutable_child(j) = child(j - 1);
        child(j)->set_position(j);
      }
      *mutable_child(i) = NULL;
    }
  }
  
  template <typename P>
  inline void btree_node<P>::remove_value(int i) {
    if (!leaf()) {
      assert(child(i + 1)->count() == 0);
      for (int j = i + 1; j < count(); ++j) {
        *mutable_child(j) = child(j + 1);
        child(j)->set_position(j);
      }
      *mutable_child(count()) = NULL;
    }
  
    set_count(count() - 1);
    for (; i < count(); ++i) {
      value_swap(i, this, i + 1);
    }
    value_destroy(i);
  }
  
  template <typename P>
  void btree_node<P>::rebalance_right_to_left(btree_node *src, int to_move) {
    assert(parent() == src->parent());
    assert(position() + 1 == src->position());
    assert(src->count() >= count());
    assert(to_move >= 1);
    assert(to_move <= src->count());
  
    // Make room in the left node for the new values.
    for (int i = 0; i < to_move; ++i) {
      value_init(i + count());
    }
  
    // Move the delimiting value to the left node and the new delimiting value
    // from the right node.
    value_swap(count(), parent(), position());
    parent()->value_swap(position(), src, to_move - 1);
  
    // Move the values from the right to the left node.
    for (int i = 1; i < to_move; ++i) {
      value_swap(count() + i, src, i - 1);
    }
    // Shift the values in the right node to their correct position.
    for (int i = to_move; i < src->count(); ++i) {
      src->value_swap(i - to_move, src, i);
    }
    for (int i = 1; i <= to_move; ++i) {
      src->value_destroy(src->count() - i);
    }
  
    if (!leaf()) {
      // Move the child pointers from the right to the left node.
      for (int i = 0; i < to_move; ++i) {
        set_child(1 + count() + i, src->child(i));
      }
      for (int i = 0; i <= src->count() - to_move; ++i) {
        assert(i + to_move <= src->max_count());
        src->set_child(i, src->child(i + to_move));
        *src->mutable_child(i + to_move) = NULL;
      }
    }
  
    // Fixup the counts on the src and dest nodes.
    set_count(count() + to_move);
    src->set_count(src->count() - to_move);
  }
  
  template <typename P>
  void btree_node<P>::rebalance_left_to_right(btree_node *dest, int to_move) {
    assert(parent() == dest->parent());
    assert(position() + 1 == dest->position());
    assert(count() >= dest->count());
    assert(to_move >= 1);
    assert(to_move <= count());
  
    // Make room in the right node for the new values.
    for (int i = 0; i < to_move; ++i) {
      dest->value_init(i + dest->count());
    }
    for (int i = dest->count() - 1; i >= 0; --i) {
      dest->value_swap(i, dest, i + to_move);
    }
  
    // Move the delimiting value to the right node and the new delimiting value
    // from the left node.
    dest->value_swap(to_move - 1, parent(), position());
    parent()->value_swap(position(), this, count() - to_move);
    value_destroy(count() - to_move);
  
    // Move the values from the left to the right node.
    for (int i = 1; i < to_move; ++i) {
      value_swap(count() - to_move + i, dest, i - 1);
      value_destroy(count() - to_move + i);
    }
  
    if (!leaf()) {
      // Move the child pointers from the left to the right node.
      for (int i = dest->count(); i >= 0; --i) {
        dest->set_child(i + to_move, dest->child(i));
        *dest->mutable_child(i) = NULL;
      }
      for (int i = 1; i <= to_move; ++i) {
        dest->set_child(i - 1, child(count() - to_move + i));
        *mutable_child(count() - to_move + i) = NULL;
      }
    }
  
    // Fixup the counts on the src and dest nodes.
    set_count(count() - to_move);
    dest->set_count(dest->count() + to_move);
  }
  
  template <typename P>
  void btree_node<P>::split(btree_node *dest, int insert_position) {
    assert(dest->count() == 0);
  
    // 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 == 0) {
      dest->set_count(count() - 1);
    } else if (insert_position == max_count()) {
      dest->set_count(0);
    } else {
      dest->set_count(count() / 2);
    }
    set_count(count() - dest->count());
    assert(count() >= 1);
  
    // Move values from the left sibling to the right sibling.
    for (int i = 0; i < dest->count(); ++i) {
      dest->value_init(i);
      value_swap(count() + i, dest, i);
      value_destroy(count() + i);
    }
  
    // The split key is the largest value in the left sibling.
    set_count(count() - 1);
    parent()->insert_value(position(), value_type());
    value_swap(count(), parent(), position());
    value_destroy(count());
    parent()->set_child(position() + 1, dest);
  
    if (!leaf()) {
      for (int i = 0; i <= dest->count(); ++i) {
        assert(child(count() + i + 1) != NULL);
        dest->set_child(i, child(count() + i + 1));
        *mutable_child(count() + i + 1) = NULL;
      }
    }
  }
  
  template <typename P>
  void btree_node<P>::merge(btree_node *src) {
    assert(parent() == src->parent());
    assert(position() + 1 == src->position());
  
    // Move the delimiting value to the left node.
    value_init(count());
    value_swap(count(), parent(), position());
  
    // Move the values from the right to the left node.
    for (int i = 0; i < src->count(); ++i) {
      value_init(1 + count() + i);
      value_swap(1 + count() + i, src, i);
      src->value_destroy(i);
    }
  
    if (!leaf()) {
      // Move the child pointers from the right to the left node.
      for (int i = 0; i <= src->count(); ++i) {
        set_child(1 + count() + i, src->child(i));
        *src->mutable_child(i) = NULL;
      }
    }
  
    // Fixup the counts on the src and dest nodes.
    set_count(1 + count() + src->count());
    src->set_count(0);
  
    // Remove the value on the parent node.
    parent()->remove_value(position());
  }
  
  template <typename P>
  void btree_node<P>::swap(btree_node *x) {
    assert(leaf() == x->leaf());
  
    // Swap the values.
    for (int i = count(); i < x->count(); ++i) {
      value_init(i);
    }
    for (int i = x->count(); i < count(); ++i) {
      x->value_init(i);
    }
    int n = std::max(count(), x->count());
    for (int i = 0; i < n; ++i) {
      value_swap(i, x, i);
    }
    for (int i = count(); i < x->count(); ++i) {
      x->value_destroy(i);
    }
    for (int i = x->count(); i < count(); ++i) {
      value_destroy(i);
    }
  
    if (!leaf()) {
      // Swap the child pointers.
      for (int i = 0; i <= n; ++i) {
        btree_swap_helper(*mutable_child(i), *x->mutable_child(i));
      }
      for (int i = 0; i <= count(); ++i) {
        x->child(i)->fields_.parent = x;
      }
      for (int i = 0; i <= x->count(); ++i) {
        child(i)->fields_.parent = this;
      }
    }
  
    // Swap the counts.
    btree_swap_helper(fields_.count, x->fields_.count);
  }
  
  ////
  // btree_iterator methods
  template <typename N, typename R, typename P>
  void btree_iterator<N, R, P>::increment_slow() {
    if (node->leaf()) {
      assert(position >= node->count());
      self_type save(*this);
      while (position == node->count() && !node->is_root()) {
        assert(node->parent()->child(node->position()) == node);
        position = node->position();
        node = node->parent();
      }
      if (position == node->count()) {
        *this = save;
      }
    } else {
      assert(position < node->count());
      node = node->child(position + 1);
      while (!node->leaf()) {
        node = node->child(0);
      }
      position = 0;
    }
  }
  
  template <typename N, typename R, typename P>
  void btree_iterator<N, R, P>::increment_by(int count) {
    while (count > 0) {
      if (node->leaf()) {
        int rest = node->count() - position;
        position += std::min(rest, count);
        count = count - rest;
        if (position < node->count()) {
          return;
        }
      } else {
        --count;
      }
      increment_slow();
    }
  }
  
  template <typename N, typename R, typename P>
  void btree_iterator<N, R, P>::decrement_slow() {
    if (node->leaf()) {
      assert(position <= -1);
      self_type save(*this);
      while (position < 0 && !node->is_root()) {
        assert(node->parent()->child(node->position()) == node);
        position = node->position() - 1;
        node = node->parent();
      }
      if (position < 0) {
        *this = save;
      }
    } else {
      assert(position >= 0);
      node = node->child(position);
      while (!node->leaf()) {
        node = node->child(node->count());
      }
      position = node->count() - 1;
    }
  }
  
  ////
  // btree methods
  template <typename P>
  btree<P>::btree(const key_compare &comp, const allocator_type &alloc)
      : key_compare(comp),
        root_(alloc, NULL) {
  }
  
  template <typename P>
  btree<P>::btree(const self_type &x)
      : key_compare(x.key_comp()),
        root_(x.internal_allocator(), NULL) {
    assign(x);
  }
  
  template <typename P> template <typename ValuePointer>
  std::pair<typename btree<P>::iterator, bool>
  btree<P>::insert_unique(const key_type &key, ValuePointer value) {
    if (empty()) {
      *mutable_root() = new_leaf_root_node(1);
    }
  
    std::pair<iterator, int> res = internal_locate(key, iterator(root(), 0));
    iterator &iter = res.first;
    if (res.second == kExactMatch) {
      // The key already exists in the tree, do nothing.
      return std::make_pair(internal_last(iter), false);
    } else if (!res.second) {
      iterator last = internal_last(iter);
      if (last.node && !compare_keys(key, last.key())) {
        // The key already exists in the tree, do nothing.
        return std::make_pair(last, false);
      }
    }
  
    return std::make_pair(internal_insert(iter, *value), true);
  }
  
  template <typename P>
  inline typename btree<P>::iterator
  btree<P>::insert_unique(iterator position, const value_type &v) {
    if (!empty()) {
      const key_type &key = params_type::key(v);
      if (position == end() || compare_keys(key, position.key())) {
        iterator prev = position;
        if (position == begin() || compare_keys((--prev).key(), key)) {
          // prev.key() < key < position.key()
          return internal_insert(position, v);
        }
      } else if (compare_keys(position.key(), key)) {
        iterator next = position;
        ++next;
        if (next == end() || compare_keys(key, next.key())) {
          // position.key() < key < next.key()
          return internal_insert(next, v);
        }
      } else {
        // position.key() == key
        return position;
      }
    }
    return insert_unique(v).first;
  }
  
  template <typename P> template <typename InputIterator>
  void btree<P>::insert_unique(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
      insert_unique(end(), *b);
    }
  }
  
  template <typename P> template <typename ValuePointer>
  typename btree<P>::iterator
  btree<P>::insert_multi(const key_type &key, ValuePointer value) {
    if (empty()) {
      *mutable_root() = new_leaf_root_node(1);
    }
  
    iterator iter = internal_upper_bound(key, iterator(root(), 0));
    if (!iter.node) {
      iter = end();
    }
    return internal_insert(iter, *value);
  }
  
  template <typename P>
  typename btree<P>::iterator
  btree<P>::insert_multi(iterator position, const value_type &v) {
    if (!empty()) {
      const key_type &key = params_type::key(v);
      if (position == end() || !compare_keys(position.key(), key)) {
        iterator prev = position;
        if (position == begin() || !compare_keys(key, (--prev).key())) {
          // prev.key() <= key <= position.key()
          return internal_insert(position, v);
        }
      } else {
        iterator next = position;
        ++next;
        if (next == end() || !compare_keys(next.key(), key)) {
          // position.key() < key <= next.key()
          return internal_insert(next, v);
        }
      }
    }
    return insert_multi(v);
  }
  
  template <typename P> template <typename InputIterator>
  void btree<P>::insert_multi(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
      insert_multi(end(), *b);
    }
  }
  
  template <typename P>
  void btree<P>::assign(const self_type &x) {
    clear();
  
    *mutable_key_comp() = x.key_comp();
    *mutable_internal_allocator() = x.internal_allocator();
  
    // Assignment can avoid key comparisons because we know the order of the
    // values is the same order we'll store them in.
    for (const_iterator iter = x.begin(); iter != x.end(); ++iter) {
      if (empty()) {
        insert_multi(*iter);
      } else {
        // If the btree is not empty, we can just insert the new value at the end
        // of the tree!
        internal_insert(end(), *iter);
      }
    }
  }
  
  template <typename P>
  typename btree<P>::iterator btree<P>::erase(iterator iter) {
    bool internal_delete = false;
    if (!iter.node->leaf()) {
      // Deletion of a value on an internal node. Swap the key with the largest
      // value of our left child. This is easy, we just decrement iter.
      iterator tmp_iter(iter--);
      assert(iter.node->leaf());
      assert(!compare_keys(tmp_iter.key(), iter.key()));
      iter.node->value_swap(iter.position, tmp_iter.node, tmp_iter.position);
      internal_delete = true;
      --*mutable_size();
    } else if (!root()->leaf()) {
      --*mutable_size();
    }
  
    // Delete the key from the leaf.
    iter.node->remove_value(iter.position);
  
    // 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.
  
    // Merge/rebalance as we walk back up the tree.
    iterator res(iter);
    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);
      if (iter.node->leaf()) {
        res = iter;
      }
      if (!merged) {
        break;
      }
      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->count()) {
      res.position = res.node->count() - 1;
      ++res;
    }
    // If we erased from an internal node, advance the iterator.
    if (internal_delete) {
      ++res;
    }
    return res;
  }
  
  template <typename P>
  int btree<P>::erase(iterator b, iterator e) {
    int count = distance(b, e);
    for (int i = 0; i < count; i++) {
      b = erase(b);
    }
    return count;
  }
  
  template <typename P>
  int btree<P>::erase_unique(const key_type &key) {
    iterator iter = internal_find_unique(key, iterator(root(), 0));
    if (!iter.node) {
      // The key doesn't exist in the tree, return nothing done.
      return 0;
    }
    erase(iter);
    return 1;
  }
  
  template <typename P>
  int btree<P>::erase_multi(const key_type &key) {
    iterator b = internal_lower_bound(key, iterator(root(), 0));
    if (!b.node) {
      // The key doesn't exist in the tree, return nothing done.
      return 0;
    }
    // Delete all of the keys between begin and upper_bound(key).
    iterator e = internal_end(
        internal_upper_bound(key, iterator(root(), 0)));
    return erase(b, e);
  }
  
  template <typename P>
  void btree<P>::clear() {
    if (root() != NULL) {
      internal_clear(root());
    }
    *mutable_root() = NULL;
  }
  
  template <typename P>
  void btree<P>::swap(self_type &x) {
    std::swap(static_cast<key_compare&>(*this), static_cast<key_compare&>(x));
    std::swap(root_, x.root_);
  }
  
  template <typename P>
  void btree<P>::verify() const {
    if (root() != NULL) {
      assert(size() == internal_verify(root(), NULL, NULL));
      assert(leftmost() == (++const_iterator(root(), -1)).node);
      assert(rightmost() == (--const_iterator(root(), root()->count())).node);
      assert(leftmost()->leaf());
      assert(rightmost()->leaf());
    } else {
      assert(size() == 0);
      assert(leftmost() == NULL);
      assert(rightmost() == NULL);
    }
  }
  
  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());
  
    // First try to make room on the node by rebalancing.
    node_type *parent = node->parent();
    if (node != root()) {
      if (node->position() > 0) {
        // Try rebalancing with our left sibling.
        node_type *left = parent->child(node->position() - 1);
        if (left->count() < left->max_count()) {
          // 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 = (left->max_count() - left->count()) /
              (1 + (insert_position < left->max_count()));
          to_move = std::max(1, to_move);
  
          if (((insert_position - to_move) >= 0) ||
              ((left->count() + to_move) < left->max_count())) {
            left->rebalance_right_to_left(node, to_move);
  
            assert(node->max_count() - node->count() == to_move);
            insert_position = insert_position - to_move;
            if (insert_position < 0) {
              insert_position = insert_position + left->count() + 1;
              node = left;
            }
  
            assert(node->count() < node->max_count());
            return;
          }
        }
      }
  
      if (node->position() < parent->count()) {
        // Try rebalancing with our right sibling.
        node_type *right = parent->child(node->position() + 1);
        if (right->count() < right->max_count()) {
          // 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 = (right->max_count() - right->count()) /
              (1 + (insert_position > 0));
          to_move = std::max(1, to_move);
  
          if ((insert_position <= (node->count() - to_move)) ||
              ((right->count() + to_move) < right->max_count())) {
            node->rebalance_left_to_right(right, to_move);
  
            if (insert_position > node->count()) {
              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.
      if (parent->count() == parent->max_count()) {
        iterator parent_iter(node->parent(), node->position());
        rebalance_or_split(&parent_iter);
      }
    } else {
      // Rebalancing not possible because this is the root node.
      if (root()->leaf()) {
        // The root node is currently a leaf node: create a new root node and set
        // the current root node as the child of the new root.
        parent = new_internal_root_node();
        parent->set_child(0, root());
        *mutable_root() = parent;
        assert(*mutable_rightmost() == parent->child(0));
      } else {
        // The root node is an internal node. We do not want to create a new root
        // node because the root node is special and holds the size of the tree
        // and a pointer to the rightmost node. So we create a new internal node
        // and move all of the items on the current root into the new node.
        parent = new_internal_node(parent);
        parent->set_child(0, parent);
        parent->swap(root());
        node = parent;
      }
    }
  
    // Split the node.
    node_type *split_node;
    if (node->leaf()) {
      split_node = new_leaf_node(parent);
      node->split(split_node, insert_position);
      if (rightmost() == node) {
        *mutable_rightmost() = split_node;
      }
    } else {
      split_node = new_internal_node(parent);
      node->split(split_node, insert_position);
    }
  
    if (insert_position > node->count()) {
      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);
    if (right->leaf()) {
      if (rightmost() == right) {
        *mutable_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() > 0) {
      // Try merging with our left sibling.
      node_type *left = parent->child(iter->node->position() - 1);
      if ((1 + left->count() + iter->node->count()) <= left->max_count()) {
        iter->position += 1 + left->count();
        merge_nodes(left, iter->node);
        iter->node = left;
        return true;
      }
    }
    if (iter->node->position() < parent->count()) {
      // Try merging with our right sibling.
      node_type *right = parent->child(iter->node->position() + 1);
      if ((1 + iter->node->count() + right->count()) <= right->max_count()) {
        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 > 0))) {
        int to_move = (right->count() - iter->node->count()) / 2;
        to_move = std::min(to_move, right->count() - 1);
        iter->node->rebalance_right_to_left(right, to_move);
        return false;
      }
    }
    if (iter->node->position() > 0) {
      // 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->count()))) {
        int to_move = (left->count() - iter->node->count()) / 2;
        to_move = std::min(to_move, left->count() - 1);
        left->rebalance_left_to_right(iter->node, to_move);
        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() = NULL;
    } else {
      node_type *child = root()->child(0);
      if (child->leaf()) {
        // The child is a leaf node so simply make it the root node in the tree.
        child->make_root();
        delete_internal_root_node();
        *mutable_root() = child;
      } else {
        // The child is an internal node. We want to keep the existing root node
        // so we move all of the values from the child node into the existing
        // (empty) root node.
        child->swap(root());
        delete_internal_node(child);
      }
    }
  }
  
  template <typename P> template <typename IterType>
  inline IterType btree<P>::internal_last(IterType iter) {
    while (iter.node && iter.position == iter.node->count()) {
      iter.position = iter.node->position();
      iter.node = iter.node->parent();
      if (iter.node->leaf()) {
        iter.node = NULL;
      }
    }
    return iter;
  }
  
  template <typename P>
  inline typename btree<P>::iterator
  btree<P>::internal_insert(iterator iter, const value_type &v) {
    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;
    }
    if (iter.node->count() == iter.node->max_count()) {
      // Make room in the leaf for the new item.
      if (iter.node->max_count() < kNodeValues) {
        // Insertion into the root where the root is smaller that 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 * iter.node->max_count()));
        iter.node->swap(root());
        delete_leaf_node(root());
        *mutable_root() = iter.node;
      } else {
        rebalance_or_split(&iter);
        ++*mutable_size();
      }
    } else if (!root()->leaf()) {
      ++*mutable_size();
    }
    iter.node->insert_value(iter.position, v);
    return iter;
  }
  
  template <typename P> template <typename IterType>
  inline std::pair<IterType, int> btree<P>::internal_locate(
      const key_type &key, IterType iter) const {
    return internal_locate_type::dispatch(key, *this, iter);
  }
  
  template <typename P> template <typename IterType>
  inline std::pair<IterType, int> btree<P>::internal_locate_plain_compare(
      const key_type &key, IterType iter) const {
    for (;;) {
      iter.position = iter.node->lower_bound(key, key_comp());
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return std::make_pair(iter, 0);
  }
  
  template <typename P> template <typename IterType>
  inline std::pair<IterType, int> btree<P>::internal_locate_compare_to(
      const key_type &key, IterType iter) const {
    for (;;) {
      int res = iter.node->lower_bound(key, key_comp());
      iter.position = res & kMatchMask;
      if (res & kExactMatch) {
        return std::make_pair(iter, static_cast<int>(kExactMatch));
      }
      if (iter.node->leaf()) {
        break;
      }
      iter.node = iter.node->child(iter.position);
    }
    return std::make_pair(iter, -kExactMatch);
  }
  
  template <typename P> template <typename IterType>
  IterType btree<P>::internal_lower_bound(
      const key_type &key, IterType iter) const {
    if (iter.node) {
      for (;;) {
        iter.position =
            iter.node->lower_bound(key, key_comp()) & kMatchMask;
        if (iter.node->leaf()) {
          break;
        }
        iter.node = iter.node->child(iter.position);
      }
      iter = internal_last(iter);
    }
    return iter;
  }
  
  template <typename P> template <typename IterType>
  IterType btree<P>::internal_upper_bound(
      const key_type &key, IterType iter) const {
    if (iter.node) {
      for (;;) {
        iter.position = iter.node->upper_bound(key, key_comp());
        if (iter.node->leaf()) {
          break;
        }
        iter.node = iter.node->child(iter.position);
      }
      iter = internal_last(iter);
    }
    return iter;
  }
  
  template <typename P> template <typename IterType>
  IterType btree<P>::internal_find_unique(
      const key_type &key, IterType iter) const {
    if (iter.node) {
      std::pair<IterType, int> res = internal_locate(key, iter);
      if (res.second == kExactMatch) {
        return res.first;
      }
      if (!res.second) {
        iter = internal_last(res.first);
        if (iter.node && !compare_keys(key, iter.key())) {
          return iter;
        }
      }
    }
    return IterType(NULL, 0);
  }
  
  template <typename P> template <typename IterType>
  IterType btree<P>::internal_find_multi(
      const key_type &key, IterType iter) const {
    if (iter.node) {
      iter = internal_lower_bound(key, iter);
      if (iter.node) {
        iter = internal_last(iter);
        if (iter.node && !compare_keys(key, iter.key())) {
          return iter;
        }
      }
    }
    return IterType(NULL, 0);
  }
  
  template <typename P>
  void btree<P>::internal_clear(node_type *node) {
    if (!node->leaf()) {
      for (int i = 0; i <= node->count(); ++i) {
        internal_clear(node->child(i));
      }
      if (node == root()) {
        delete_internal_root_node();
      } else {
        delete_internal_node(node);
      }
    } else {
      delete_leaf_node(node);
    }
  }
  
  template <typename P>
  void btree<P>::internal_dump(
      std::ostream &os, const node_type *node, int level) const {
    for (int i = 0; i < node->count(); ++i) {
      if (!node->leaf()) {
        internal_dump(os, node->child(i), level + 1);
      }
      for (int j = 0; j < level; ++j) {
        os << "  ";
      }
      os << node->key(i) << " [" << level << "]\n";
    }
    if (!node->leaf()) {
      internal_dump(os, node->child(node->count()), level + 1);
    }
  }
  
  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(0), *lo));
    }
    if (hi) {
      assert(!compare_keys(*hi, node->key(node->count() - 1)));
    }
    for (int i = 1; i < node->count(); ++i) {
      assert(!compare_keys(node->key(i), node->key(i - 1)));
    }
    int count = node->count();
    if (!node->leaf()) {
      for (int i = 0; i <= node->count(); ++i) {
        assert(node->child(i) != NULL);
        assert(node->child(i)->parent() == node);
        assert(node->child(i)->position() == i);
        count += internal_verify(
            node->child(i),
            (i == 0) ? lo : &node->key(i - 1),
            (i == node->count()) ? hi : &node->key(i));
      }
    }
    return count;
  }
  
  } // namespace btree
  
  #endif  // UTIL_BTREE_BTREE_H__
  