Lecture on Red Black Trees

Introduction

• Topics Covered: Need for Red Black Trees, Comparison with AVL Trees, Properties, Examples.

Need for Red Black Tree

• Data Structures: Array, Linked List, Stack, Queue, Tree.
• Tree: Organizes data efficiently for quick search, insertion, update, deletion.
• Binary Search Tree (BST): Organizes nodes such that left subtree < node < right subtree.
• BST Pros & Cons: Average search time is O(log n), worst case O(n).
• Balanced BST: Ensures balanced tree for O(log n) operations.
• Skewed Trees: Right or left skewed BST can lead to O(n) search time.
• Self-Balancing Trees: Red Black Tree and AVL Tree ensure balance.

Red Black Tree vs AVL Tree

• AVL Trees: Strictly height balanced, ensures O(log n) time complexities for all operations.
• Red Black Trees: Roughly height balanced, fewer rotations required.
• Rotation: Maximum 2 rotations in Red Black Trees vs. potentially many in AVL Trees.
• Recoloring: Nodes can be recolored to maintain balance; max two rotations might be required.
• Use Case: Prefer AVL for faster search, Red Black Tree for frequent insertion/deletion.

Properties of Red Black Trees

1. Binary Search Tree: Follows BST properties.
2. Node Color: Each node is red or black.
3. Root Property: Root is always black.
4. Leaf Property: All leaves (NIL) are black.
5. Red Property: Red nodes can’t have red children (no red-red parent-child).
6. Black Property: Every path from a node to its descendant NIL nodes has the same number of black nodes.
7. Structure: Nodes have color bit, key/data, left and right pointers.
8. Relative Height Balance: Longest path from root to any leaf is no more than twice the length of the shortest path.

Examples & Analysis

Example 1

• BST: No (Root is red).

Example 2

• All Black Nodes: Yes, but not Red Black Tree (Unequal black nodes in some paths).

Example 3

• Invalid BST: No (All Red Property violated).

Example 4

• Valid Red Black Tree: Yes (Balanced paths of black nodes).

Example 5

• Possible Red Black Tree Modifications: Changing node colors to maintain properties.

Example 6

• Red Node Near Root: Validity depends on maintaining both red and black properties.

Example 7

• Student Question: Analyze a given tree for Red Black Tree properties.

Conclusion

• Choice of Tree: Depends on specific operations and performance requirements.
• Next Steps: Perform insertions in Red Black Trees.

Homework

• Analyze given tree structure to determine if it’s a Red Black Tree or not.