AVL tree rotations in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Single Rotations in AVL Trees

What is the main purpose of a single right rotation in an AVL tree?

A To swap the left and right children of a node without changing the tree height

B To balance a right-heavy subtree by moving the right child up and the root down to the left

C To balance a left-heavy subtree by moving the left child up and the root down to the right

D To remove a node from the tree without affecting balance

💡 Hint

Think about which side is heavy and how the rotation moves nodes to restore balance.

A single right rotation is used when a node's left subtree is heavier. It moves the left child up to the root position and the original root down to the right, restoring balance.

🧠 Conceptual

intermediate

Identifying When to Use Double Rotations

In which situation is a double rotation (left-right or right-left) necessary in an AVL tree?

A When the left child of a node is right-heavy or the right child is left-heavy

B When the tree is perfectly balanced and no rotations are needed

C When the root node has no children

D When the tree has only one node

💡 Hint

Consider the imbalance direction of the child subtree relative to its parent.

Double rotations are needed when a subtree is unbalanced in a zig-zag pattern, such as a left child having a right-heavy subtree or a right child having a left-heavy subtree. This requires two rotations to fix.

🔍 Analysis

advanced

Result of a Left-Right Rotation

Given an AVL tree where node 10 has a left child 5, and node 5 has a right child 7, what is the structure of the subtree after performing the correct rotation to balance it?

Data Structures Theory

Initial subtree:
    10
   /
  5
   \
    7

A 7 becomes the new root, with 5 as its left child and 10 as its right child

B 5 remains the root, with 7 as its left child and 10 as its right child

C 10 remains the root, with 7 as its left child and 5 as its right child

D 7 becomes the new root, with 10 as its left child and 5 as its right child

💡 Hint

Think about the left-right double rotation steps: first left rotation on 5, then right rotation on 10.

The left-right rotation first rotates node 5 left around 7, then rotates node 10 right around 7. This makes 7 the new root, with 5 as left child and 10 as right child.

❓ Comparison

advanced

Difference Between Single and Double Rotations

Which statement correctly describes the difference between single and double rotations in AVL trees?

A Single rotations are used only for right-heavy trees; double rotations only for left-heavy trees

B Single rotations fix simple imbalances on one side; double rotations fix complex imbalances involving opposite child directions

C Single rotations always increase tree height; double rotations always decrease tree height

D Single rotations remove nodes; double rotations add nodes

💡 Hint

Consider the imbalance patterns each rotation type addresses.

Single rotations handle simple cases where imbalance is on one side. Double rotations handle zig-zag cases where the child subtree is heavy in the opposite direction.

❓ Reasoning

expert

Height Changes After Rotations

After performing a single left rotation on an AVL tree node, what happens to the height of the subtree rooted at that node?

A The height becomes zero, making the subtree empty

B The height increases by one, causing further imbalance

C The height remains the same, but balance factors change

D The height decreases by one, restoring balance

💡 Hint

Think about how rotations affect subtree height and balance.

A single left rotation reduces the height of the heavier right subtree, decreasing the overall height of the subtree by one and restoring balance.

Practice

1. What is the main purpose of rotations in an AVL tree?

easy

A. To keep the tree balanced for faster search and update operations

B. To delete nodes from the tree

C. To increase the height of the tree

D. To sort the nodes in ascending order

Solution

1. Step 1: Understand AVL tree balance

   AVL trees maintain balance by ensuring the height difference between left and right subtrees is at most 1.

2. Step 2: Role of rotations

   Rotations adjust the tree structure to restore balance after insertions or deletions, improving operation speed.

3. Final Answer:

   To keep the tree balanced for faster search and update operations -> Option A

4. Quick Check:

   AVL rotations = balance tree = faster operations [OK]

Hint: Rotations fix balance to keep operations fast [OK]

Common Mistakes:

• Thinking rotations delete nodes
• Believing rotations sort nodes
• Assuming rotations increase tree height

2. Which of the following is the correct syntax to perform a right rotation on a node x in an AVL tree?

easy

A. rightRotate(x)

B. x.rightRotate()

C. rotateRight(x)

D. rotate(x, 'right')

Solution

1. Step 1: Identify common function naming

   In AVL tree implementations, the function to rotate right is commonly named rotateRight(node).

2. Step 2: Check options

   rotateRight(x) matches the common syntax. rightRotate(x) and x.rightRotate() are less standard, and rotate(x, 'right') is a generic call not typical in AVL code.

3. Final Answer:

   rotateRight(x) -> Option C

4. Quick Check:

   Right rotation function = rotateRight(node) [OK]

Hint: Look for 'rotateRight' as standard right rotation function name [OK]

Common Mistakes:

• Using method calls on node objects incorrectly
• Confusing rightRotate with rotateRight
• Using generic rotate function without direction

3. Given the following AVL tree insertion sequence: Insert 10, then 20, then 30. Which rotation will be performed to balance the tree?

medium

A. Left rotation

B. Right rotation

C. Left-Right rotation

D. Right-Left rotation

Solution

1. Step 1: Analyze insertion order and imbalance

   Inserting 10, then 20, then 30 creates a right-heavy chain (10 -> 20 -> 30), causing imbalance at 10.

2. Step 2: Determine rotation type

   This is a Right-Right case, fixed by a single left rotation at node 10.

3. Final Answer:

   Left rotation -> Option A

4. Quick Check:

   Right-heavy imbalance = left rotation [OK]

Hint: Right-heavy chain = left rotation to balance [OK]

Common Mistakes:

• Choosing right rotation for right-heavy imbalance
• Confusing Left-Right with Right-Right cases
• Ignoring the order of insertions

4. You implemented a left-right rotation but the AVL tree remains unbalanced after insertion. What is the most likely error?

medium

A. Performed rotations in wrong order

B. Used right rotation instead of left rotation

C. Inserted duplicate keys

D. Forgot to update node heights after rotation

Solution

1. Step 1: Understand rotation steps

   Left-right rotation requires two rotations and updating node heights to maintain AVL properties.

2. Step 2: Identify common mistake

   Not updating heights after rotations causes incorrect balance checks, leaving tree unbalanced.

3. Final Answer:

   Forgot to update node heights after rotation -> Option D

4. Quick Check:

   Update heights after rotations to keep balance [OK]

Hint: Always update heights after rotations [OK]

Common Mistakes:

• Doing rotations in wrong order
• Mixing up left and right rotations
• Ignoring height updates

5. After inserting nodes 30, 20, 25 into an empty AVL tree, which rotation(s) will balance the tree?

hard

A. Single left rotation at 20

B. Left-Right rotation at 30

C. Single right rotation at 30

D. Right-Left rotation at 20

Solution

1. Step 1: Analyze insertion sequence and imbalance

   Inserting 30, then 20, then 25 creates a left-right case: 30 has left child 20, which has right child 25.

2. Step 2: Identify correct rotation

   Left-right imbalance requires a left rotation on 20 followed by a right rotation on 30, called a left-right rotation.

3. Final Answer:

   Left-Right rotation at 30 -> Option B

4. Quick Check:

   Left-right case = left-right rotation [OK]

Hint: Left child with right-heavy subtree = left-right rotation [OK]

Common Mistakes:

• Choosing single rotations instead of double
• Confusing left-right with right-left cases
• Rotating wrong nodes