Complete Binary Tree

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. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are actually arborescences. A binary tree may thus be also called a bifurcating arborescence’a term which actually appears in some very old programming books, 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. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special case of an ordered K-ary tree, where k is 2. There is however a subtle difference between the binary tree data structure as defined here and the notions from graph theory or as K-ary tree is usually defined. As defined here, a binary tree node having a left child but no right child is not the same as a node having a right child but no left child, whereas an ordered/plane tree (or arborescence) from graph theory cannot tell these cases apart, and neither does k-ary as usually understood as using a list of children. An actual generalization of binary tree would have to discern, for example, a case like having a first and third, but no second child; the trie data structure is actually the more appropriate generalization in this respect. 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. 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. In mathematics, what is termed binary tree can vary significantly from author to author. Some use the definition commonly used computer science, but others define it as every non-leaf having exactly two children and don’t necessarily order (as left/right) the children either.