<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 {
    int max_ele(vector<int>& arr, int i, int j){
        int mx = 0;
        for(int k=i; k<=j; k++)  mx = max(mx, arr[k]);
        return mx;
    }
    int solve(int l, int r, vector<int>& arr, vector<vector<int>>& dp){
        if(r == l)  return 0;
        if(dp[l][r] != -1)  return dp[l][r];

        int ans = INT_MAX;
        for(int k=l; k<r; k++){
            int mx1 = max_ele(arr, l, k);
            int mx2 = max_ele(arr, k+1, r);
            ans = min(ans, mx1*mx2 + solve(l, k, arr, dp) + solve(k+1, r, arr, dp));
        }
        return dp[l][r] = ans;
    }
public:
    int mctFromLeafValues(vector<int>& arr) {
        if(arr.size() == 2)  return arr[0]*arr[1];
        vector<vector<int>> dp(41, vector<int>(41, -1));
        return solve(0, arr.size()-1, arr, dp);
    }
};
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