Smallest Subtree with all the Deepest Nodes

Smallest Subtree with all the Deepest Nodes

What is this problem about?

The "Smallest Subtree with all the Deepest Nodes" problem is a fascinating tree-based challenge often encountered in interviews at Meta and Salesforce. You are given a binary tree, and your goal is to find the node that is the root of the smallest subtree containing all the nodes that are at the maximum depth of the tree.

Depth is defined as the distance from the root. If there's only one deepest node, that node itself is the answer. If there are multiple nodes at the maximum depth (for example, at the far left and far right of the bottom level), you must find their lowest common ancestor. This node is the "root" of the smallest subtree that encompasses all those deepest points.

Why is this asked in interviews?

This interview question is popular because it tests multiple tree traversal concepts at once. To solve it, a candidate must demonstrate knowledge of depth-first search (DFS) or breadth-first search (BFS) to identify the maximum depth. Furthermore, it requires an understanding of the Lowest Common Ancestor (LCA) pattern. It checks if you can effectively pass information (like depth) back up the recursive stack, which is a fundamental skill for advanced tree problems.

Algorithmic pattern used

The most efficient algorithmic pattern for this problem is Depth-First Search (DFS) with a post-order traversal. As you visit each node, you recursively determine the depth of its left and right subtrees. If the depths of both subtrees are equal and they both lead to the maximum depth found in the tree, then the current node is a candidate for the smallest subtree root. This approach allows you to solve the problem in a single pass ($O(N)$ time complexity).

Example explanation

Consider a tree where the root is 3. Its left child is 5 and right is 1.

• Node 5 has children 6 and 2.
• Node 2 has children 7 and 4.
• Max depth is reached at nodes 7 and 4. When we perform DFS:
• Node 2 sees that its left child (7) and right child (4) both have the same maximum depth. It returns itself as the root of the subtree containing 7 and 4.
• Node 5 sees that its right child (2) contains the deepest nodes, while its left child (6) does not reach the max depth. It passes the result from node 2 up to node 3.
• Node 3 sees that its left side contains the deepest nodes and its right side does not. The final answer is node 2.

Common mistakes candidates make

One common mistake is performing multiple passes—one to find the max depth, another to find all deepest nodes, and a third to find their LCA. While this works, it's not optimal and can be harder to implement correctly. Another mistake is confusing "depth" (distance from root) with "height" (distance to leaf), which can lead to incorrect logic when comparing subtrees. Finally, candidates sometimes fail to return both the depth and the node as a pair in their recursive function, making the implementation clunky.

Interview preparation tip

Mastering the "Smallest Subtree with all the Deepest Nodes coding problem" requires a strong grasp of recursion. A great tip is to practice returning multiple values from a recursive function (like a tuple or a custom object containing both the current max depth and the current best ancestor node). This "bottom-up" information flow is a very common interview pattern for tree and graph problems. Also, review the standard Lowest Common Ancestor problem, as this is essentially a variation of it.