1 Trees Examples of Tree structure Definition of trees Binary tree Height of tree Tree traversals Finding max Finding sum2Trees You have seen that using linked lists you can represent an ordered collection of values without using arrays. Although linked lists require more memory space than arrays ( as they have to store address at each node), they have definite advantages over arrays. Insertion and deletion of items can be carried out with out involving considerable movement of data. The ordering relationship amongst a set of values is obtained through use of pointers. However, we need not restrict ourselves to only linear structures. In this chapter we shall extend the use of pointers to define a non-linear structure to model hierarchical relationships, such as a family tree. In such a tree, we have links moving from an ancestor to a parent, and links moving from the parent to children. We have many other examples of tree-structured hierarchies. Directory Hierarchies: In computers, files are stored in directories that form a tree. The top level directory represents the root. It has many subdirectories and3files. The subdirectories would have further set of subdirectories. Organization charts: In a company a number of vice presidents report to a president. Each VP would have a set of general managers, each GM having his/her own set of specific managers and so on. Biological classifications: Starting from living being at the root, such a tree can branch off to mammals, birds, marine life etc. Game Trees: All games which require only mental effort would always have number of possible options at any position of the game. For each position, there would be number of counter moves. The repetitive pattern results in what is known a game tree. Tree as a data structure • A tree is a data structure that is made of nodes and pointers, much like a linked list. The difference between them lies in how they are organized: • The top node in the tree is called the root and all other nodes branch off from this one.4• Every node in the tree can have some number of children. Each child node can in turn be the parent node to its children and so on. • Child nodes can have links only from a single parent. • Any node higher up than the parent is called an ancestor node. • Nodes having no children are called leaves. • Any node which is neither a root, nor a leaf is called an interior node. • The height of a tree is defined to be the length of the longest path from the root to a leaf in that tree ( including the path to root) • A common example of a tree structure is the binary tree. Binary Trees5Definition: A binary tree is a tree in which each node can have maximum two children. Thus each node can have no child, one child or two children. The pointers help us to identify whether it is a left child or a right child. Application of a Binary tree Before we define any formal algorithms, let us look at one possible application of a binary tree. Consider a set of numbers: 25,63,13, 72,18,32,59,67. Suppose we store these numbers in individual nodes of a singly linked list. To search for a particular item we have to go through the list, and maybe we have to go to the end of the list as well. Thus if there were n numbers, our search complexity would be O(n). Is it because the numbers are not in any particular sequence? Now suppose we order these numbers: 13,18,25,32,59,63,67,72. and store these in another linked list. What would be the search complexity now? You may be surprised to discover that it is still O(n). You simply cannot apply binary search on a linked list with O(log n) complexity. You still have to go6through each link to locate a particular number. So a linear linked structure is not helping us at all. Let us see if we can improve the situation by storing the data using a binary tree structure. Consider the following binary tree where the numbers have been stored in a specific order. The value at any node is more than the values stored in the left-child nodes, and less than the values stored in the right-child nodes. 59 18 67 13 32 63 72 257With this arrangement any search is taking at most 4 steps. For larger set of numbers, if we can come up with a good tree arrangement than the search time can be reduced dramatically. Examples of binary trees: root root8 • The following are NOT binary trees:9Definitions: • tree, then n1 is the parent of n2 and n2 is the left or right child of n1. • The level of a node in a binary tree: - The root of the tree has level 0 - The level of any other node in the tree is one more than the level of its parent. root Level 0 Level 1 Level 2 Level 310Full Binary Tree How many nodes? Level 0 : 1 node ( height 1) Level 1: 2 nodes ( height 2) Level 3 : 4 nodes (height 3) Level 3: 8 nodes (height 4) Total number of nodes n = 2h – 1 ( maximum) h = log ( n+1)45 24 76 14 32 61 87 8 20 27 3756 67 81 9411Implementation • A binary tree has a natural implementation in linked storage. A tree is referenced with a pointer to its root. • Recursive definition of a binary tree: A binary tree is either - Empty, or - A node (called root) together with two binary trees (called left subtree and the right subtree of the root) • Each node of a binary tree has both left and right subtrees which can be reached with pointers: struct tree_node{ int data; struct tree_node *left_child; struct tree_node *right_child; }; left_child data right_child12Note the recursive definition of trees. A tree is a node with structure that contains more trees. We have actually a tree located at each node of a tree. Traversal of Binary Trees Linked lists are traversed sequentially from first node to the last node. However, there is no such natural linear order for the nodes of a tree. Different orderings are possible for traversing a binary tree. Every node in the tree is a root for the subtree that it points to. There are three common traversals for binary trees: • Preorder • Inorder • Postorder These names are chosen according to the sequence in which the root node and its children are visited. Suppose there are only 3 nodes in the tree having the following arrangement:13 − With inorder traversal the order is left-child, root node, right-child − With preorder