Tree: A Non-linear Data Structure

Tree: A Non-linear Data Structure Programming and Data Structure

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Binary Tree • A new data structure • Binary means two. • Definition • A binary tree is either empty, or it consists of a node called the root, together with two binary trees called the left subtree and the right subtree. Root Left subtree Right subtree Programming and Data Structure

Examples of Binary Trees Programming and Data Structure

Not Binary Trees Programming and Data Structure

How to Implement a Binary Tree? • Two pointers in every node (left and right). struct nd { int element; struct nd *lptr; struct nd *rptr; }; typedef nd node; node *root; /* Will point to the root of the tree */ Programming and Data Structure

An Example a • Create the tree a = (node *) malloc (sizeof (node)); b = (node *) malloc (sizeof (node)); c = (node *) malloc (sizeof (node)); d = (node *) malloc (sizeof (node)); a->element = 10; a->lptr = b; a->rptr = c; b->element = 5; b->lptr = NULL; b->rptr = NULL; c->element = 20; c->lptr = d; c->rptr = NULL; d->element = 15; d->lptr = NULL; d->rptr = NULL; root = a; 10 b c 5 20 15 d Programming and Data Structure

Traversal of Binary Trees • In many applications, it is required to move through all the nodes of a binary tree, visiting each node in turn. • For n nodes, there exists n! different orders. • Three traversal orders are most common: • Inorder traversal • Preorder traversal • Postorder traversal Programming and Data Structure

Inorder Traversal • Recursively, perform the following three steps: • Visit the left subtree. • Visit the root. • Visit the right subtree. LEFT-ROOT-RIGHT Programming and Data Structure

Example:: inorder traversal 10 20 30 10 20 30 40 25 50 60 20 10 30 40 20 25 10 50 30 60 Programming and Data Structure

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Preorder Traversal • Recursively, perform the following three steps: • Visit the root. • Visit the left subtree. • Visit the right subtree. ROOT-LEFT-RIGHT Programming and Data Structure

10 20 30 40 25 50 60 Example:: preorder traversal 10 20 30 10 20 30 10 20 40 25 30 50 60 Programming and Data Structure

a b c d e f g h i j k a b d . g . c e h i j k f Programming and Data Structure

Postorder Traversal • Recursively, perform the following three steps: • Visit the left subtree. • Visit the right subtree. • Visit the root. LEFT-RIGHT-ROOT Programming and Data Structure

Example:: postorder traversal 10 20 30 10 20 30 40 25 50 60 20 30 10 40 25 20 50 60 30 10 Programming and Data Structure

a b c d e f g h i j k . g d . b h j k i e f c a Programming and Data Structure

Implementations void inorder (node *root) { if (root != NULL) { inorder (root->left); printf (“%d “, root->element); inorder (root->right); } } void preorder (node *root) { if (root != NULL) { printf (“%d “, root->element); inorder (root->left); inorder (root->right); } } void postorder (node *root) { if (root != NULL) { inorder (root->left); inorder (root->right); printf (“%d “, root->element); } } Programming and Data Structure

Graph :: another important data structure • Definition • A graph G={V,E} consists of a set of vertices V, and a set of edges E which connect pairs of vertices. • If the edges are undirected, G is called an undirected graph. • If at least one edge in G is directed, it is called a directed graph. Programming and Data Structure

How to represent a graph in a program? • Many ways • Adjacency matrix • Incidence matrix, etc. Programming and Data Structure

Adjacency Matrix e1 e5 1 2 6 e2 e4 e6 e7 e3 e8 3 4 5 . 1 2 3 4 5 6 1 0 1 1 0 0 0 2 1 0 0 1 1 1 3 1 0 0 1 0 0 4 0 1 1 0 1 0 5 0 1 0 1 0 1 6 0 1 0 0 1 0 Programming and Data Structure

Incidence Matrix e1 e5 1 2 6 e2 e4 e6 e7 e3 e8 3 4 5 . e1 e2 e3 e4 e5 e6 e7 e8 1 1 1 0 0 0 0 0 0 2 1 0 0 1 1 1 0 0 3 0 1 1 0 0 0 0 0 4 0 0 1 1 0 0 0 1 5 0 0 0 0 0 1 1 1 6 0 0 0 0 1 0 1 0 Programming and Data Structure

Tree: A Non-linear Data Structure