4.4. Binary Trees¶
4.4.1. Introduction¶
Tree structures enable efficient access and efficient update to large collections of data. Binary trees in particular are widely used and relatively easy to implement. But binary trees are useful for many things besides searching. Just a few examples of applications that trees can speed up include prioritizing jobs, describing mathematical expressions and the syntactic elements of computer programs, or organizing the information needed to drive data compression algorithms.
This chapter covers terminology used for discussing binary trees, tree traversals, approaches to implementing tree nodes, and various examples of binary trees.
4.4.2. Definitions and Properties¶
A binary tree is made up of a finite set of elements called nodes. This set either is empty or consists of a node called the root together with two binary trees, called the left and right subtrees, which are disjoint from each other and from the root. (Disjoint means that they have no nodes in common.) The roots of these subtrees are children of the root. There is an edge from a node to each of its children, and a node is said to be the parent of its children.
If \(n_1, n_2, ..., n_k\) is a sequence of nodes in the tree such that \(n_i\) is the parent of \(n_i+1\) for \(1 \leq i < k\), then this sequence is called a path from \(n_1\) to \(n_k\). The length of the path is \(k-1\). If there is a path from node \(R\) to node \(M\), then \(R\) is an ancestor of \(M\), and \(M\) is a descendant of \(R\). Thus, all nodes in the tree are descendants of the root of the tree, while the root is the ancestor of all nodes. The depth of a node \(M\) in the tree is the length of the path from the root of the tree to \(M\). The height of a tree is the depth of the deepest node in the tree. All nodes of depth \(d\) are at level \(d\) in the tree. The root is the only node at level 0, and its depth is 0. A leaf node is any node that has two empty children. An internal node is any node that has at least one non-empty child.
Figure 4.4.1: A binary tree. Node \(A\) is the root. Nodes \(B\) and \(C\) are \(A\)’s children. Nodes \(B\) and \(D\) together form a subtree. Node \(B\) has two children: Its left child is the empty tree and its right child is \(D\). Nodes \(A\), \(C\), and \(E\) are ancestors of \(G\). Nodes \(D\), \(E\), and \(F\) make up level 2 of the tree; node \(A\) is at level 0. The edges from \(A\) to \(C\) to \(E\) to \(G\) form a path of length 3. Nodes \(D\), \(G\), \(H\), and \(I\) are leaves. Nodes \(A\), \(B\), \(C\), \(E\), and \(F\) are internal nodes. The depth of \(I\) is 3. The height of this tree is 3.
Figure 4.4.2: Two different binary trees. (a) A binary tree whose root has a non-empty left child. (b) A binary tree whose root has a non-empty right child. (c) The binary tree of (a) with the missing right child made explicit. (d) The binary tree of (b) with the missing left child made explicit.
Figure 4.4.1 illustrates the various terms used to identify parts of a binary tree. Figure 4.4.2 illustrates an important point regarding the structure of binary trees. Because all binary tree nodes have two children (one or both of which might be empty), the two binary trees of Figure 4.4.2 are not the same.
Two restricted forms of binary tree are sufficiently important to warrant special names. Each node in a full binary tree is either (1) an internal node with exactly two non-empty children or (2) a leaf. A complete binary tree has a restricted shape obtained by starting at the root and filling the tree by levels from left to right. In a complete binary tree of height \(d\), all levels except possibly level \(d\) are completely full. The bottom level has its nodes filled in from the left side.
Figure 4.4.3: Examples of full and complete binary trees.
Figure 4.4.3 illustrates the differences between full and complete binary trees. 1 There is no particular relationship between these two tree shapes; that is, the tree of Figure 4.4.3 (a) is full but not complete while the tree of Figure 4.4.3 (b) is complete but not full. The heap data structure is an example of a complete binary tree. The Huffman coding tree is an example of a full binary tree.
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While these definitions for full and complete binary tree are the ones most commonly used, they are not universal. Because the common meaning of the words “full” and “complete” are quite similar, there is little that you can do to distinguish between them other than to memorize the definitions. Here is a memory aid that you might find useful: “Complete” is a wider word than “full”, and complete binary trees tend to be wider than full binary trees because each level of a complete binary tree is as wide as possible.
4.4.3. Binary Tree as a Recursive Data Structure¶
A recursive data structure is a data structure that is partially composed of smaller or simpler instances of the same data structure. For example, linked lists and binary trees can be viewed as recursive data structures. A list is a recursive data structure because a list can be defined as either (1) an empty list or (2) a node followed by a list. A binary tree is typically defined as (1) an empty tree or (2) a node pointing to two binary trees, one its left child and the other one its right child.
The recursive relationships used to define a structure provide a natural model for any recursive algorithm on the structure.
