of adjacency list. In relational databases, nodes are typically represented as table rows, with indexed row IDs facilitating pointers between parents and children. Nodes can also be stored as items in an array, with relationships between them determined by their positions in the array (as in a binary heap). A binary tree can be implemented as a list of lists: the head of a list (the value of the first term) is the left child (subtree), while the tail (the list of second and subsequent terms) is the right child (subtree). This can be modified to allow values as well, as in Lisp S-expressions, where the head (value of first term) is the value of the node, the head of the tail (value of second term) is the left child, and the tail of the tail (list of third and subsequent terms) is the right child. Ordered trees can be naturally encoded by finite sequences, for example with natural numbers. Type theory As an abstract data type, the abstract tree type with values of some type is defined, using the abstract forest type (list of trees), by the functions: value: → children: → nil: () → node: × → with the axioms: value(node(, )) = children(node(, )) = In terms of type theory, a tree is an inductive type defined by the constructors (empty forest) and (tree with root node with given value and children). Mathematical terminology Viewed as a whole, a tree data structure is an ordered tree, generally with values attached to each node. Concretely, it is (if required to be non-empty): A rooted tree with the "away from root" direction (a more narrow term is an "arborescence"), meaning: A directed graph, whose underlying undirected graph is a tree (any two vertices are connected by exactly one simple path), with a distinguished root (one vertex is designated as the root), which determines the direction on the edges (arrows point away from the root; given an edge, the node that the edge points from is called the parent and the node that the edge points to is called the child), together with: an ordering on the child nodes of a given node, and a value (of some data type) at each node. Often trees have a fixed (more properly, bounded) branching factor (outdegree), particularly always having two child nodes (possibly empty, hence at most two non-empty child nodes), hence