Virtual University CS301 Data Structure Short Lecture 19 || CS301 Short Lectures

Summary

In this tutorial, the speaker explains binary search trees (BST) and AVL trees, focusing on their construction, insertion logic, and balancing techniques. The video walks through building a degenerate binary search tree and introduces the concept of AVL trees—self-balancing trees that ensure optimal search times by maintaining a height difference of no more than 1 between left and right subtrees. Practical examples and theoretical concepts, like the use of the `const` keyword in programming, are included, making it an engaging resource for programming students and enthusiasts looking to understand tree data structures and algorithms.

Takeaways

• 😀 The `const` keyword in programming ensures that a pointer’s value cannot be modified, making it constant and immutable.
• 😀 A Binary Search Tree (BST) is a tree structure where each node's left child holds a value smaller than the node, and the right child holds a larger value.
• 😀 A degenerate binary search tree occurs when all nodes are placed on one side, resembling a linked list, which reduces search efficiency.
• 😀 When inserting nodes into a binary search tree, each node is compared with the current node, determining whether it should be placed on the left or right side.
• 😀 The AVL Tree is a self-balancing binary search tree that maintains balance by ensuring the height difference between the left and right subtrees is at most one.
• 😀 The balance factor of a node in an AVL tree is the difference between the heights of its left and right subtrees. A balance factor outside the range of -1 to +1 indicates the need for rebalancing.
• 😀 If an AVL tree becomes unbalanced, rotations (left, right, left-right, right-left) are performed to restore balance.
• 😀 Left and right rotations in AVL trees are used to balance the tree when one subtree becomes too heavy on one side.
• 😀 AVL trees are introduced by the scientists Adelson-Velsky and Landis, and they are known for their ability to keep the tree balanced for optimal search performance.
• 😀 The AVL tree ensures that its balance is maintained through continuous adjustments, providing efficient search, insert, and delete operations.

Q & A

What is a Binary Search Tree (BST)?

A Binary Search Tree (BST) is a type of binary tree where each node follows two main properties: all nodes to the left of a given node have smaller values, and all nodes to the right of a given node have larger values. This allows efficient searching, insertion, and deletion operations.

How are nodes inserted into a Binary Search Tree?

In a Binary Search Tree, nodes are inserted by comparing the value of the node with the current node. If the new value is smaller, it moves to the left; if it is larger, it moves to the right. This process continues until the correct position for the new node is found.

What are the key properties of an AVL tree?

An AVL tree is a self-balancing binary search tree. The key property is that the difference in height between the left and right subtrees of any node must be at most 1. If the balance factor exceeds 1 or -1, the tree must be rebalanced through rotations.

What is a balance factor in an AVL tree?

The balance factor in an AVL tree is the difference in height between the left and right subtrees of a node. A balance factor of 0, 1, or -1 indicates that the tree is balanced. If the balance factor is greater than 1 or less than -1, the tree is unbalanced and needs to be rotated.

What is the purpose of rotations in AVL trees?

Rotations are used in AVL trees to restore balance when the balance factor of a node becomes too large or too small. There are four types of rotations: left rotation, right rotation, left-right rotation, and right-left rotation, each used based on the type of imbalance.

How is a tree balanced using a right rotation?

A right rotation is used when a tree is left-heavy (i.e., the balance factor is greater than 1). In a right rotation, the left child of the root becomes the new root, and the original root becomes the right child of the new root.

What does it mean when an AVL tree has a balance factor of 1, 0, or -1?

If an AVL tree has a balance factor of 1, 0, or -1, it indicates that the tree is balanced. A balance factor of 1 means the left subtree is one level higher, 0 means both subtrees have equal height, and -1 means the right subtree is one level higher.

What is the role of the height in AVL trees?

The height of a tree is the number of edges along the longest path from the root to a leaf. In AVL trees, the height is crucial for determining the balance factor and deciding when rotations are needed to maintain balance.

What are the types of imbalance in AVL trees and how are they fixed?

There are four types of imbalance in AVL trees: left-left (LL), right-right (RR), left-right (LR), and right-left (RL). Each type of imbalance is fixed using the appropriate rotation: a single left or right rotation for LL or RR, and a double rotation for LR or RL.

What is the difference between a Binary Search Tree (BST) and an AVL tree?

The primary difference between a Binary Search Tree (BST) and an AVL tree is that an AVL tree is self-balancing. In an AVL tree, the heights of the left and right subtrees of every node must differ by at most one, while a BST does not enforce any balance condition.