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.