Trees

18CS32 - DATA STRUCTURES AND APPLICATIONS | Module-4 VTU Notes

VTU | 18CS32 | Module - 4

Trees, Binary Trees, and Binary Search Trees

This summary provides an overview of trees, binary trees, binary search trees, their operations, and applications, as covered in the "Data Structures and Applications" course.

Trees: Terminology:

Trees are hierarchical structures composed of nodes. Essential terms include root, parent, child, sibling, leaf, height, depth, and level.

Binary Trees: Definition and Properties:

Binary trees are trees where each node has at most two children. Binary trees possess unique properties that influence their structure and traversal.

Array and Linked Representation of Binary Trees:

Binary trees can be represented using arrays or linked structures. Arrays maintain structural integrity, while linked structures allow dynamic sizing.

Binary Tree Traversals: Inorder, Postorder, Preorder:

Tree traversals dictate the order in which nodes are visited. Inorder, postorder, and preorder traversals have distinct sequences, influencing how nodes are processed.

Additional Binary Tree Operations:

Operations include calculating the height, counting nodes, determining the number of leaf nodes, and checking if a binary tree is balanced.

Threaded Binary Trees:

Threaded binary trees optimize traversals by adding thread pointers, reducing recursive overhead.

Binary Search Trees (BST): Definition, Insertion, Deletion:

BSTs are binary trees with specific ordering properties. Insertion maintains the order, while deletion requires reorganizing nodes to retain the binary search property.

Binary Tree Traversal in BST, Searching:

Inorder traversal of a BST yields sorted elements. Searching is efficient due to the binary search property.

Application of Trees: Expression Evaluation:

Trees are used to evaluate expressions by converting infix expressions to postfix and then using a binary expression tree for efficient evaluation.

Examples:

1. Binary Tree Representation: Illustrate the array and linked representation of a binary tree with nodes and their relationships.

2. Binary Tree Traversal: Provide examples of inorder, postorder, and preorder traversals on a binary tree.

3. Binary Search Tree: Showcase insertion and deletion operations on a BST, maintaining the binary search property.

4. Expression Evaluation: Demonstrate the conversion of infix expressions to postfix notation and evaluate the expression using a binary expression tree.

Conclusion:

Trees, binary trees, and binary search trees are fundamental data structures with a wide range of applications. Understanding their terminology, properties, operations, and applications equips students to efficiently organize data and solve complex problems. Through examples and practical exercises, students gain hands-on experience in manipulating trees and applying their concepts in real-world scenarios. This knowledge forms the foundation for further exploration of advanced data structures and algorithmic techniques.