# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement Design a stack that supports push (x), pop(), top(), peekMax(), and popMax(). The popMax operation must remove and return the maximum element; if there are multiple maximums, remove the most recently pushed one. Implement two approaches: (A) an amortized O( 1) solution using auxiliary structures, and (B) an O(log n) popMax solution using a balanced tree or heap plus linked nodes. Explain how you will handle duplicates, empty-structure edge cases, and concurrency concerns for a high-throughput system. Provide time/space complexities and brief test cases. ## Recommended Approach Choose traversal based on the required output. DFS is natural for subtree computations, reconstruction, and range pruning; BFS is natural for level order or side views. Keep per-depth or per-position state when the output depends on columns, rows, or depths. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity Most tree traversals are O(n) time and O(h) recursion stack for DFS or O(w) queue space for BFS, where h is height and w is maximum width. ## Edge Cases and Tests Empty tree, one node, skewed tree, duplicate values when reconstruction assumes uniqueness, deep recursion, and tie-breaking for same row/column nodes.