Is this a full time job?

Question:

What is the worst-case time complexity of searching for an element in a balanced Binary Search Tree (BST) containing n nodes?

Options:

    
• A) O(1)
    
• B) O(log n)
    
• C) O(n)
    
• D) O(n log n)

Correct Answer: B) O(log n)

Detailed Explanation:

In a balanced Binary Search Tree (such as an AVL tree or a Red-Black tree), the height of the tree is guaranteed to be proportional to log2(n).

During a search operation:

    
• The algorithm compares the target key with the root node.
    
• If the target does not match, the search proceeds to either the left or right child depending on whether the target is smaller or larger.
    
• Because the tree is balanced, each comparison discards roughly half of the remaining search space.
    
• The maximum number of comparisons needed corresponds to the maximum depth (height) of the tree, which is O(log n).

Analysis of Incorrect Options:

    
• Option A (O(1)): Constant time is typical for direct array indexing or average-case Hash Table lookups, not tree traversals.
    
• Option C (O(n)): Linear time occurs in an unbalanced (skewed) Binary Search Tree where nodes form a single path similar to a linked list. However, the question specifies a balanced tree.
    
• Option D (O(n log n)): This time complexity is typical for efficient comparison-based sorting algorithms (such as Merge Sort), not for searching a single element.

Key Takeaway: Balancing ensures that tree operations remain logarithmic, avoiding the performance degradation seen in worst-case degenerate trees.