Minimum Cost Tree From Leaf Values

Minimum Cost Tree From Leaf Values

1. What is this problem about?

The Minimum Cost Tree From Leaf Values problem asks you to construct a binary tree from an array of integers that represent the leaf nodes in an in-order traversal. Each non-leaf node has a value equal to the product of the largest leaf in its left subtree and the largest leaf in its right subtree. The goal is to build the tree such that the sum of the values of all non-leaf nodes is minimized. This is a unique problem because the "cost" is determined by the maximum values of the subtrees you combine.

2. Why is this asked in interviews?

This problem is a favorite at Bloomberg and Google because it can be solved in multiple ways: Dynamic Programming, Greedy, or using a Monotonic Stack. The Minimum Cost Tree From Leaf Values interview question evaluates whether a candidate can move beyond the obvious DP approach to find a more optimal Greedy or Stack-based solution. It tests the ability to recognize that this problem is equivalent to repeatedly merging adjacent elements.

3. Algorithmic pattern used

The most efficient pattern is the Monotonic Stack (Greedy). To minimize the product of two largest leaves, we want to pair the smallest available leaf with its smaller neighbor as early as possible. We maintain a stack of values in decreasing order. When we find an element larger than the top of the stack, we know the top element is a local minimum. We "merge" it with the smaller of its two neighbors (the one below it in the stack or the current element) and add that product to our total. This "Monotonic Stack, Greedy interview pattern" reduces the complexity from $O(N^3)$ (DP) to $O(N)$.

4. Example explanation

Consider leaves = [6, 2, 4].

• Option 1: Merge (6, 2) first. Max(6, 2) = 6. Product = 12. Remaining: [6, 4]. Merge (6, 4). Max(6, 4) = 6. Product = 24. Total = 12 + 24 = 36.
• Option 2: Merge (2, 4) first. Max(2, 4) = 4. Product = 8. Remaining: [6, 4]. Merge (6, 4). Max(6, 4) = 6. Product = 24. Total = 8 + 24 = 32. The minimum cost is 32.

5. Common mistakes candidates make

In the Minimum Cost Tree From Leaf Values coding problem, many candidates jump straight to DP ($O(N^3)$), which is correct but not the most efficient. A common mistake in the greedy logic is not realizing that you must always merge the current smallest element with its immediate smaller neighbor to guarantee the overall minimum. Some also struggle with the monotonic stack implementation, specifically handling the elements left in the stack after the initial pass.

6. Interview preparation tip

Learn to recognize "merging" problems. If a problem involves combining adjacent elements to form a tree or a single value, it can often be solved with a Monotonic Stack or a Greedy approach. This "Stack-based interview pattern" is a powerful tool for optimizing problems that otherwise look like they require expensive dynamic programming.