Data Structures and Algorithms Lecture (BinaryTrees) Instructor: Quratulain

Introduction Tree is a non-linear structure. In computer science Tree is a ADT of hierarchical structure. Tree is divided into levels from root to leaf nodes.

Tree Terminology The height of a tree is the maximum level in the tree.

Terminology Node, branch, root, indegree, outdegree, leaf, parent, child, siblings, ancestor, descendent, path, level, height, depth, subtree

Binary Tree Each node has 0, 1 and 2 childs Each node of a binary tree defines a left and a right subtree. Each subtree is itself a tree.

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. Height of a Binary Tree The height of a binary tree is the length of the longest path from the root to a leaf node. Let T N be the subtree with root N and T L and T R be the roots of the left and right subtrees of N. Then -1if T N is empty 1+max( height(T L ), height(T R ))if T N not empty height(N) = height(T N ) = {

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. Height of a Binary Tree (concluded) Degenerate binary tree

Types Binary Trees Complete binary tree: A complete binary tree of height h has all possible nodes through level h-1, and some nodes on depth h exist left to right with no gaps. Full binary tree: A tree in which every node other than the leaves has two children Perfect binary tree: A full binary tree in which all leaves are at the same level

Binary tree Total number of nodes in complete binary tree is from 2 h – 1 to 2 h +1 The height of complete binary tree h=(log 2 n)

Some valid Binary Search Trees

Some invalid Binary Search Trees

Application of Binary Trees For two-way decisions at each point in a process. Then the number of comparison could be reduced. to path finding, connected components. Application of Sorting Application of searching Expression tree

Linklist and Arrays for binary tree Store in array using formula: 2n+1 for left child 2n+2 for right child

Binary Trees: Traversals There are three classic ways to traverse a tree: NLR, LNR and LRN

Preorder (NLR) Traversal preorder(node) { if (node is not null) process(node) preorder(node->left) preorder(node->right) }

Inorder (LNR) Traversal inorder(node) { if (node is not null) inorder(node->left) process(node) inorder(node->right) }

Postorder (LRN) Traversal postorder(node) { if (node is not null) inorder(node->left) inorder(node->right) process(node) }

Traversing Binary Search Trees Preorder: Postorder: Inorder:

Delete to Binary Search Trees The deletion algorithm is more complicated, because after a non-leaf node is deleted, the "hole" in the structure needs to be filled by a leaf node. There are three possibilities: ◦ To delete a leaf node (no children): disconnect it. ◦ To delete a node with one child: bypass the node and directly connect to the child. ◦ To delete a node with two children: find the smallest node in its right subtree (or the largest node in its left subtree), copy the minimum value into the info field of the "node to be deleted" and then delete the minimum node instead, which can only have a right child, so the situation becomes one of the above two.

Binary Search Trees vs. Arrays Same O(log 2 N) search Better insertion time: O(log 2 N) vs. O(N) Better deletion What is worse? O(N) BST requires more space - 2 references for each data element

Huffman Codes Binary trees can be used in an interesting way to construct minimal length encodings for messages when the frequency of letters used in the messages is known. A special kind of binary tree, called a Huffman coding tree is used to accomplish this. (Chapter 8 for detail)

Huffman Codes Huffman is a coding algorithm presented by David Huffman in It's an algorithm which works with integer length codes. Huffman is the best option because it's optimal. The position of the symbol depends on its probability. Then it assigns a code based on its position in the tree. The codes have the prefix property and are instantaneously decodable thus they are well suited for compression and decompression.

Example suppose we know that the frequency of occurrences for six letters in a message are as given below: To build the Huffman tree, we sort the frequencies into increasing order (4, 5, 7, 8, 12, 29). Then we choose the two smallest values, 4 and 5, and construct a binary tree with labeled edges: E29 I5 N7 P12 S4 T8

Next, we replace the two smallest values 4 and 5 with their sum, getting a new sequence, (7, 8, 9, 12, 29). We again take the two smallest values and construct a labeled binary tree. We now have the frequencies (15, 9, 12, 29) which must be sorted into (9, 12, 15, 29) and the two lowest are selected once again. Now, we combine the two lowest which are 15 and 21 to give the tree.

The two remaining frequencies, 36 and 29, are now combined into the final tree. Notice that it does not make any difference which one is placed as the left subtree and which in the right subtree.

From this final tree, we find the encoding for this alphabet Using this code, a message like SENT would be coded as If the message had been coded in the "normal" way, each letter would have required 3 bits. The entire message is 65 characters long so 195 bits would be needed to code the message (3*65). Using the Huffman code, the message requires 1*29+4*5+3*12+3*7+4*4+3*8=146 bits. This code can be applied to the English Language by using average frequency counts for the letters. E1 I0110 P010 N000 S0111 T001