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Unit 3: AVL Trees & M-Way Search Trees — Data Structure Notes

AVL Trees: The Need for Balance

The Problem: A BST fed sorted data (10, 20, 30, 40) degenerates into a linked list. Search becomes O(n) instead of O(log n).

BST vs AVL — Comparison

Feature	BST	AVL Tree
Balance	No guarantee	Always balanced (BF ∈ {-1, 0, +1})
Search	O(log n) avg, O(n) worst	O(log n) always
Insert	O(log n) avg, O(n) worst	O(log n) + rotation overhead
Delete	O(log n) avg, O(n) worst	O(log n) + rotation overhead
Height	Can be n (skewed)	Always ≤ 1.44 log₂(n)
Overhead	None	Must maintain heights + rotations
Best For	Random data	When worst-case guarantee needed

AVL Balance Factor

Balance Factor (BF) = Height(Left Subtree) - Height(Right Subtree)

BF Value	Meaning	Action
0	Perfectly balanced	No action needed
+1	Left-heavy by 1	Acceptable
-1	Right-heavy by 1	Acceptable
+2 or more	Left-heavy violation	Rotation needed
-2 or less	Right-heavy violation	Rotation needed

The 4 Rotation Cases

Case	Imbalance Pattern	Fix	Steps
LL (Left-Left)	Inserted in left subtree of left child	Single Right Rotation	Lift left child up
RR (Right-Right)	Inserted in right subtree of right child	Single Left Rotation	Lift right child up
LR (Left-Right)	Inserted in right subtree of left child	Left Rotation then Right Rotation	Double rotation
RL (Right-Left)	Inserted in left subtree of right child	Right Rotation then Left Rotation	Double rotation

Worked Example: AVL Insertion with Rotation

Insert sequence: 30, 20, 10

Step 1: Insert 30 → [30] (BF = 0) Step 2: Insert 20 → 30 has BF = +1 (acceptable) Step 3: Insert 10 → 30 has BF = +2 → LL Case!

Before rotation:

    30 (BF=+2)            20 (BF=0)
   /                     /  \
  20 (BF=+1)    →      10    30
 /
10

After Right Rotation: Tree is balanced with height 1 instead of 2.

M-Way Search Trees

An M-Way Search Tree (order M) allows each node to have up to M children and M-1 keys.

M-Way Properties

Property	Binary Tree	M-Way Tree (Order M)
Max Children	2	M
Max Keys per Node	1	M - 1
Height (n items)	O(log₂ n)	O(log_M n)
Disk Reads (n=10⁶)	~20	~3-4 (M=100)
Best For	In-memory data	Disk-based storage

Node Structure: [ P0, K1, P1, K2, P2 ... Km-1, Pm ]

Keys (K): Sorted values inside the node.
Pointers (P): Links to sub-trees.
All values in P0's subtree < K1 < all values in P1's subtree < K2 ...