Data Structures and Algorithm Analysis - Computer Science at ...

5Binary TreesThe list representations of Chapter 4 have a fundamental limitation: Either searchor insert can be made efficient, but not both at the same time. Tree structurespermit both efficient access and update to large collections of data. Binary trees inparticular are widely used and relatively easy to implement. But binary trees areuseful for many things besides searching. Just a few examples of applications thattrees can speed up include prioritizing jobs, describing mathematical expressionsand the syntactic elements of computer programs, or organizing the informationneeded to drive data compression algorithms.This chapter begins by presenting definitions and some key properties of binarytrees. Section 5.2 discusses how to process all nodes of the binary tree in anorganized manner. Section 5.3 presents various methods for implementing binarytrees and their nodes. Sections 5.4 through 5.6 present three examples of binarytrees used in specific applications: the Binary Search Tree (BST) for implementingdictionaries, heaps for implementing priority queues, and Huffman coding trees fortext compression. The BST, heap, and Huffman coding tree each have distinctivestructural features that affect their implementation and use.5.1 Definitions and PropertiesA binary tree is made up of a finite set of elements called nodes. This set eitheris empty or consists of a node called the root together with two binary trees, calledthe 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 subtreesare children of the root. There is an edge from a node to each of its children, anda 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 ofn i+1 for 1 ≤ i < k, then this sequence is called a path from n 1 to n k . The lengthof the path is k − 1. If there is a path from node R to node M, then R is an ancestorof M, and M is a descendant of R. Thus, all nodes in the tree are descendants of the145