L11 tree

April 12, 2023 Programming and Data Structure 1

Tree: A Non-linear Data Structure


Background

• Linked lists use dynamic memory allocation, and hence enjoys some advantages over static representations using arrays.

• A problem– Searching an ordinary linked list for a given

element.• Time complexity O(n).

– Can the nodes in a linked list be rearranged so that we can search in O(nlog2n) time?


Illustration

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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

Leftsubtree

Rightsubtree


Examples of Binary Trees


Not Binary Trees


Binary Search Trees

• A particular form of binary tree suitable for searching.

• Definition– A binary search tree is a binary tree that is either

empty or in which each node contains a key that satisfies the following conditions:

• All keys (if any) in the left subtree of the root precede the key in the root.

• The key in the root precedes all keys (if any) in its right subtree.

• The left and right subtrees of the root are again binary search trees.


Examples

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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 */


An Example

• 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;

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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


Inorder Traversal

• Recursively, perform the following three steps:– Visit the left subtree.

– Visit the root.

– Visit the right subtree.

LEFT-ROOT-RIGHT


Example:: inorder traversal

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Preorder Traversal

• Recursively, perform the following three steps:– Visit the root.

– Visit the left subtree.


ROOT-LEFT-RIGHT


Example:: preorder traversal

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Postorder Traversal

• Recursively, perform the following three steps:– Visit the left subtree.


– Visit the root.

LEFT-RIGHT-ROOT


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Example:: postorder traversal

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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); }}


A case study :: Expression Tree

*

cba

-+

+

d e

Represents the expression: (a + b) * (c – (d + e))

Preorder traversal :: * + a b - c + d e

Postorder traversal :: a b + c d e + - *


Binary Search Tree (Revisited)

• Given a binary search tree, how to write a program to search for a given element?

• Easy to write a recursive program. int search (node *root, int key) { if (root != NULL) { if (root->element == key) return (1); else if (root->element > key) search (root->lptr, key); else search (root->rptr, key); } }


Some Points to Note

• The following operations are a little complex and are not discussed here.– Inserting a node into a binary search tree.

– Deleting a node from a binary search tree.


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.


How to represent a graph in a program?

• Many ways– Adjacency matrix

– Incidence matrix, etc.


Adjacency Matrix

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. 1 2 3 4 5 61 0 1 1 0 0 02 1 0 0 1 1 13 1 0 0 1 0 04 0 1 1 0 1 05 0 1 0 1 0 16 0 1 0 0 1 0


Incidence Matrix

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. e1 e2 e3 e4 e5 e6 e7 e81 1 1 0 0 0 0 0 02 1 0 0 1 1 1 0 03 0 1 1 0 0 0 0 04 0 0 1 1 0 0 0 15 0 0 0 0 0 1 1 16 0 0 0 0 1 0 1 0

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