The base problem is easy enough, just DFS and track depth.
Clarify the tree structure and traversal order, then propose a depth-first pre-order traversal that tracks the maximum depth and the first node encountered at that depth. Walk through a small example to verify the logic, then analyze time and space complexity.
Pro tip: Mention that pre-order traversal naturally visits nodes in the required order, so the first node at the maximum depth is automatically the correct answer. Also note that an iterative solution avoids recursion depth limits for very deep trees.
Confirm whether the tree is binary or n-ary, and whether 'greatest depth' means maximum number of edges from the root. Ask if the tree can be empty or have a single node.
Select a depth-first pre-order traversal (root, then children left-to-right) to ensure the first node at the maximum depth is found. Alternatively, use BFS level-order but track the first node at the last level.
Recursively or iteratively traverse the tree, passing the current depth. Maintain global variables for max depth and the corresponding node. Update only when a strictly greater depth is found.
Trace the algorithm on a sample tree with multiple deepest nodes to demonstrate that the first one in pre-order is returned. Show how the max depth and node are updated.
State O(n) time and O(h) space for recursion (or O(n) for iterative stack). Discuss edge cases: empty tree, single node, skewed tree, and multiple deepest nodes.
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