Maximum Product of Splitted Binary Tree

Maximum Product of Splitted Binary Tree

1. What is this problem about?

The Maximum Product of Splitted Binary Tree interview question is a classic tree-based optimization problem. You are given the root of a binary tree. Your task is to split the tree into two separate subtrees by removing exactly one edge, such that the product of the sums of the values in the two subtrees is maximized.

Essentially, you are looking for an edge that divides the total sum of all node values into two parts, $S_1$ and $S_2$, where $S_1 * S_2$ is as large as possible. Since $S_1 + S_2 = \text{TotalSum}$, the problem is equivalent to finding a subtree sum $S_1$ that is as close to $\text{TotalSum} / 2$ as possible.

2. Why is this asked in interviews?

Companies like Microsoft and Amazon ask the Maximum Product of Splitted Binary Tree coding problem to evaluate a candidate's proficiency in tree traversal and recursive problem-solving. It tests whether you can efficiently calculate subtree sums without redundant computations. It also assesses your ability to handle large numerical outputs (often requiring modulo arithmetic) and your understanding of the mathematical relationship between the sum and product of two numbers.

3. Algorithmic pattern used

The Depth-First Search, Binary Tree, Tree interview pattern is the standard approach.

1. Perform a first DFS pass to calculate the total sum of all nodes in the tree.
2. Perform a second DFS pass (or use a list to store results from the first pass) to calculate the sum of every possible subtree.
3. For each subtree sum $S_{sub}$, calculate the product $P = S_{sub} * (\text{TotalSum} - S_{sub})$.
4. Track the maximum product found during the traversal.

4. Example explanation

Consider a simple tree: 1 /
2 3 /
4 5

• Total Sum = 1 + 2 + 3 + 4 + 5 = 15.
• Possible splits:
  1. Remove edge (1-2): Subtree sums are 2 and (15-2) = 13. Product = 26.
  2. Remove edge (1-3): Subtree sums are (3+4+5) = 12 and (15-12) = 3. Product = 36.
  3. Remove edge (3-4): Subtree sums are 4 and (15-4) = 11. Product = 44.
  4. Remove edge (3-5): Subtree sums are 5 and (15-5) = 10. Product = 50.

The maximum product is 50.

5. Common mistakes candidates make

A frequent mistake in the Maximum Product of Splitted Binary Tree coding problem is failing to use the correct data types for the product. Since the total sum can be large, the product can easily exceed the range of a 32-bit integer. Candidates also sometimes perform redundant calculations, such as re-traversing the tree for every edge, leading to an $O(N^2)$ solution instead of the optimal $O(N)$. Finally, ensure you apply the modulo only at the very end to avoid incorrect comparisons during the optimization phase.

6. Interview preparation tip

When dealing with tree problems that involve subtree properties, think about how you can use Post-order Traversal (DFS). In Post-order, you process the children before the parent, which is perfect for aggregating sums or heights from the bottom up. Practice storing these intermediate results in a list or a hash map to avoid multiple passes.