<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:
    vector<vector<pair<int,int>>> dp;

    pair<int,int> solve(vector<int>& arr,int l,int r){
        if(l==r){
            return {arr[l],0};
        }

        pair<int,int> temp={-1,-1};
        if(dp[l][r]!=temp){
            return dp[l][r];
        }
        int max_leaf=INT_MIN;
        int min_prod_sum=INT_MAX;
        
        for(int i=l;i<r;i++){
            auto left=solve(arr,l,i);
            auto right=solve(arr,i+1,r);

            int max_leaf_subtree=max(left.first,right.first);
            max_leaf=max(max_leaf,max_leaf_subtree);

            min_prod_sum=min(min_prod_sum,left.second+right.second+left.first*right.first);
        }
        return dp[l][r]={max_leaf,min_prod_sum};
    }
    int mctFromLeafValues(vector<int>& arr) {
        int n=arr.size();
        pair<int,int> temp={-1,-1};
        dp.resize(45,vector<pair<int,int>>(45, {-1,-1}));
        pair<int,int> ans=solve(arr,0,n-1);
        return ans.second;
    }
};
12800