What is the time complexity of the optimal DFS with two-value return solution for the House Robber III problem on a tree with n nodes?

medium🪤 Complexity Trap Q13 of Q15

Tree: Depth-First Search - House Robber III (On Tree)

AO(n^2) due to checking grandchildren nodes explicitly

BO(n) but with extra O(n) space for memoization arrays

CO(n) because each node is visited once in DFS

DO(n log n) because of recursive calls and combining results

Step-by-Step Solution

Step 1: Identify DFS traversal cost
The algorithm visits each node exactly once in a postorder DFS traversal.
Step 2: Analyze operations per node
At each node, constant time operations are done: combining left and right child results, no repeated work.
Final Answer:
Option C -> Option C
Quick Check:
Linear traversal of n nodes -> O(n) time [OK]

Quick Trick: DFS visits each node once, no repeated subproblems [OK]

Common Mistakes:

MISTAKES

Trap Explanation:

PITFALL

Thinking grandchildren checks cause repeated work leads to O(n^2) misconception.

Interviewer Note:

CONTEXT

Checks if candidate understands DFS traversal cost and avoids overestimating complexity.

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