<p>Given an array <code>arr</code> of positive integers, consider all binary trees such that:</p>

<ul>
	<li>Each node has either <code>0</code> or <code>2</code> children;</li>
	<li>The values of <code>arr</code> correspond to the values of each <strong>leaf</strong> in an in-order traversal of the tree.</li>
	<li>The value of each non-leaf node is equal to the product of the largest leaf value in its left and right subtree, respectively.</li>
</ul>

<p>Among all possible binary trees considered, return <em>the smallest possible sum of the values of each non-leaf node</em>. It is guaranteed this sum fits into a <strong>32-bit</strong> integer.</p>

<p>A node is a <strong>leaf</strong> if and only if it has zero children.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/10/tree1.jpg" style="width: 500px; height: 169px;" />
<pre>
<strong>Input:</strong> arr = [6,2,4]
<strong>Output:</strong> 32
<strong>Explanation:</strong> There are two possible trees shown.
The first has a non-leaf node sum 36, and the second has non-leaf node sum 32.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/10/tree2.jpg" style="width: 224px; height: 145px;" />
<pre>
<strong>Input:</strong> arr = [4,11]
<strong>Output:</strong> 44
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= arr.length &lt;= 40</code></li>
	<li><code>1 &lt;= arr[i] &lt;= 15</code></li>
	<li>It is guaranteed that the answer fits into a <strong>32-bit</strong> signed integer (i.e., it is less than 2<sup>31</sup>).</li>
</ul>

class Solution {
public:
    int mctFromLeafValues(vector<int>& arr) {
        int n = arr.size();
        dp.resize(n, vector<int>(n, -1));
        return mctRec(arr, 0, n-1);
    }

    int mctRec(vector<int>& arr, int s, int e){
        if(s == e) return 0;
        if(dp[s][e] != -1) return dp[s][e];
        int res = INT_MAX;
        for(int i = s; i < e; i++){
            int left = mctRec(arr, s, i);
            int right = mctRec(arr, i+1, e);
            int leftMax = INT_MIN, rightMax = INT_MIN;
            for(int j = s; j <= i; j++) leftMax = max(leftMax, arr[j]);
            for(int j = i+1; j <= e; j++) rightMax = max(rightMax, arr[j]);
            res = min(res, left + right + leftMax * rightMax);
        }
        return dp[s][e] = res;
    }

private:
    vector<vector<int>> dp;
};
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