AVL tree rotations in Data Structures Theory - Step-by-Step Execution

Concept Flow - AVL tree rotations

Insert Node

↓

Check Balance

↓

Is balance factor > 1 or < -1?

No→Tree Balanced, Stop

Yes↓

Identify Rotation Type

↓

Perform Rotation

↓

Tree Balanced, Stop

When a node is inserted, check the balance factor. If unbalanced, identify and perform the correct rotation to restore balance.

Execution Sample

Data Structures Theory

Insert 30
Insert 20
Insert 10
Check balance
Perform Right Rotation

Inserting nodes causes imbalance; a right rotation fixes the left-left case.

Analysis Table

Step	Action	Balance Factor at Node 30	Rotation Needed?	Rotation Type	Tree State
1	Insert 30	0	No	None	30
2	Insert 20	1	No	None	30 / 20
3	Insert 10	2	Yes	Right Rotation	Before Rotation: 30 / 20 / 10
4	Perform Right Rotation	0	No	None	After Rotation: 20 / \ 10 30
5	Check balance	0	No	None	Balanced Tree

💡 Tree is balanced after rotation; no further action needed.

State Tracker

Variable	Start	After Insert 30	After Insert 20	After Insert 10	After Rotation	Final
Balance Factor at Node 30	0	0	1	2	0	0
Tree Structure	Empty	30	30 with left child 20	30 with left child 20 and left-left child 10	20 root with children 10 and 30	Balanced AVL Tree

Key Insights - 3 Insights

Why do we perform a right rotation after inserting 10?

Because the balance factor at node 30 became 2 (greater than 1), indicating a left-left heavy tree, which requires a right rotation to balance (see execution_table step 3 and 4).

What happens if we don't perform any rotation when balance factor is 2?

The tree remains unbalanced, which can degrade search performance. The execution_table shows the tree is unbalanced at step 3, so rotation is necessary.

How do we know which rotation to perform?

By checking the balance factor and the side of insertion. Left-left case needs right rotation; right-right case needs left rotation. This is shown in the 'Rotation Type' column in execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the balance factor at node 30?

A 2

B 1

C 0

D -1

💡 Hint

Check the 'Balance Factor at Node 30' column in execution_table row for step 3.

At which step does the tree become balanced after rotation?

A Step 3

B Step 5

C Step 4

D Step 2

💡 Hint

Look at the 'Tree State' and 'Rotation Needed?' columns in execution_table rows for steps 4 and 5.

If we inserted 40 instead of 10 after 30 and 20, which rotation would likely be needed?

A Right Rotation

B Left Rotation

C Right-Left Rotation

D No Rotation

💡 Hint

Consider the balance factor direction and insertion side; see key_moments about rotation types.

Concept Snapshot

AVL Tree Rotations:
- Insert node
- Check balance factor (height difference)
- If balance factor >1 or <-1, tree is unbalanced
- Identify case: Left-Left, Right-Right, Left-Right, Right-Left
- Perform corresponding rotation (single or double)
- Tree becomes balanced again

Full Transcript

AVL tree rotations keep the tree balanced after insertions. When a new node is added, we check the balance factor of affected nodes. If the balance factor is more than 1 or less than -1, the tree is unbalanced. Depending on the insertion pattern, we perform a right rotation, left rotation, or double rotations to restore balance. For example, inserting nodes 30, 20, then 10 causes a left-left imbalance at node 30, fixed by a right rotation. This process ensures the tree remains balanced for efficient searching.