BST balancing problem in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Why is balancing important in a Binary Search Tree?

Consider a Binary Search Tree (BST). Why is it important to keep the tree balanced?

A To keep the height of the tree low, improving search, insert, and delete operations.

B To allow the tree to store duplicate values efficiently.

C To make sure all nodes have two children.

D To ensure the tree uses the least memory possible.

💡 Hint

Think about how the height of the tree affects the time it takes to find a value.

Balancing a BST keeps its height low, which means operations like search, insert, and delete can be done quickly, typically in O(log n) time. An unbalanced tree can degrade to a linked list with O(n) time.

📋 Factual

intermediate

What is the maximum height of a balanced BST with n nodes?

What is the maximum height of a balanced Binary Search Tree containing n nodes?

A log₂(n)

B n/2

C n log n

D n

💡 Hint

Balanced trees aim for the smallest possible height relative to the number of nodes.

A balanced BST has height proportional to log₂(n), which ensures efficient operations.

🔍 Analysis

advanced

Identify the imbalance type in this BST after insertion

After inserting a new node into a BST, the tree becomes unbalanced. The imbalance occurs when a node's left child's right subtree is taller than its left subtree. What type of imbalance is this?

A Left-Left (LL) imbalance

B Left-Right (LR) imbalance

C Right-Right (RR) imbalance

D Right-Left (RL) imbalance

💡 Hint

Think about which subtree of the left child is causing the imbalance.

When the left child's right subtree is taller, it is a Left-Right (LR) imbalance, which requires a double rotation to fix.

❓ Comparison

advanced

Compare AVL and Red-Black Trees in balancing

Which statement correctly compares AVL trees and Red-Black trees regarding balancing?

A AVL trees are less strictly balanced than Red-Black trees, so they have faster insertions.

B Red-Black trees are more strictly balanced than AVL trees, leading to faster searches.

C AVL trees are more strictly balanced than Red-Black trees, resulting in faster lookups but slower insertions and deletions.

D Both AVL and Red-Black trees have the same balancing rules and performance characteristics.

💡 Hint

Consider how strict the balancing rules are and how that affects operation speeds.

AVL trees maintain a stricter balance, which makes searches faster but insertions and deletions slower due to more rotations. Red-Black trees allow more imbalance, making insertions and deletions faster but searches slightly slower.

❓ Reasoning

expert

Determining the number of rotations needed to balance a BST

Given a BST that becomes unbalanced after inserting a node causing a Right-Left (RL) imbalance at a certain node, how many rotations are needed to restore balance?

A One single rotation

B Three rotations: two left rotations and one right rotation

C Two rotations: first right rotation on the right child, then left rotation on the unbalanced node

D Two rotations: first left rotation on the right child, then right rotation on the unbalanced node

💡 Hint

Think about the standard fix for RL imbalance in AVL trees.

For a Right-Left imbalance, the fix is a double rotation: first a left rotation on the right child, then a right rotation on the unbalanced node.