In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a triple (L, S, R), where L and R are binary trees or the empty set and S is a singleton set.[1] Some authors allow the binary tree to be the empty set as well.[2]
From a graph theory perspective, binary (and K-ary) trees as defined here are actually arborescences.[3] A binary tree may thus be also called a bifurcating arborescence[3]—a term which actually appears in some very old programming books,[4] before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree.
In computing, binary trees are seldom used solely for their structure. Much more typical is to define a labeling function on the nodes, which associates some value to each node.[9] Binary trees labelled this way are used to implement binary search trees and binary heaps, and are used for efficient searching and sorting. The designation of non-root nodes as left or right child even when there is only one child present matters in some of these applications, in particular it is significant in binary search trees.[10] In mathematics, what is termed binary tree can vary significantly from author to author. Some use the defini