This problem asks for constructing a binary tree from an array of positive integers while minimizing the sum of all non-leaf nodes. Using a state transition dynamic programming approach, you can systematically compute subarray solutions and combine them to get the global minimum. Careful handling of subarray maxima and memoization ensures correct results within feasible time limits.
Problem Statement
Given an array of positive integers representing leaf node values, construct all possible binary trees where each leaf corresponds to an element in the array. Every non-leaf node's value is the product of the largest leaf values in its left and right subtrees. Compute the minimum possible sum of all non-leaf node values across all valid trees.
Return the smallest sum of non-leaf node values for any binary tree built from the array. Each node is a leaf if and only if it has no children, and the array length is guaranteed to be between 2 and 40, with each element between 1 and 15.
Examples
Example 1
Input: arr = [6,2,4]
Output: 32
There are two possible trees shown. The first has a non-leaf node sum 36, and the second has non-leaf node sum 32.
Example 2
Input: arr = [4,11]
Output: 44
Example details omitted.
Constraints
- 2 <= arr.length <= 40
- 1 <= arr[i] <= 15
- It is guaranteed that the answer fits into a 32-bit signed integer (i.e., it is less than 231).
Solution Approach
Dynamic Programming on Subarrays
Define dp(i, j) as the minimum sum of non-leaf nodes for subarray arr[i..j]. For each subarray, try all possible partitions k where i <= k < j, and recursively compute dp(i, k) + dp(k+1, j) + max(arr[i..k]) * max(arr[k+1..j]). Store results in a memo table to avoid recomputation.
Monotonic Stack Optimization
Instead of full DP, use a monotonic decreasing stack to greedily combine smaller leaves first. Push each leaf onto the stack, and whenever the current leaf is greater than the stack top, pop and accumulate the product with the smaller neighbor. This produces the minimum non-leaf sum efficiently in O(n) time.
Handling Edge Cases and Small Arrays
For arrays of length two, the solution is simply arr[0] * arr[1]. Ensure that when computing subarray maxima or combining stack elements, indices are correctly managed to prevent overcounting. These details prevent subtle DP failures and stack miscalculations.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The DP approach has O(n^3) time due to considering all subarrays and all partitions, and O(n^2) space for memoization. The monotonic stack approach reduces time to O(n) with O(n) space. Both respect the problem's constraint that n <= 40.
What Interviewers Usually Probe
- Do you consider overlapping subproblems for the subarray DP?
- Can you reduce the O(n^3) DP with a greedy or stack-based method?
- How do you maintain maximum leaf values efficiently during subarray merges?
Common Pitfalls or Variants
Common pitfalls
- Forgetting to correctly compute max(arr[i..k]) and max(arr[k+1..j]) in DP can inflate the sum.
- Incorrect stack handling can merge the wrong neighbors, giving a larger non-leaf sum.
- Not memoizing subarray results leads to TLE on larger arrays up to 40 elements.
Follow-up variants
- Minimizing non-leaf sum when node value is the sum of maximum leaves instead of product.
- Finding maximum non-leaf sum instead of minimum for the same leaf array.
- Allowing leaves to appear multiple times, requiring adjusted DP state.
How GhostInterview Helps
- Automatically sets up dp(i,j) states and computes subarray minima for you.
- Provides monotonic stack simulations to quickly verify greedy merges and sum calculations.
- Highlights edge cases and common DP failures like subarray max miscalculations.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What pattern is used to solve Minimum Cost Tree From Leaf Values?
The problem uses state transition dynamic programming on subarrays and can be optimized with a monotonic stack for greedy merging.
Can this problem be solved faster than standard DP?
Yes, using a monotonic stack reduces time complexity from O(n^3) to O(n) while preserving the minimum sum computation.
How do you handle arrays with only two elements?
Simply compute the product of the two elements as the single non-leaf node value.
Why is memoization important in DP approach?
Memoization prevents recalculating subarray results, avoiding exponential recomputation and ensuring feasibility for n up to 40.
What common mistake leads to incorrect minimum sum?
Miscalculating subarray maximums or incorrectly merging stack elements often produces a sum larger than the true minimum.
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