<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;
    int f(vector<int>& nList,int start ,int end,int& maxi){
        if(start >= end){
            maxi = nList[start];
            return 0;
        }
        
        if(start + 1 == end){
            maxi = max(nList[start] , nList[end]);
            return nList[start] * nList[end];
        }
        if(dp[start][end].first != -1 ){
            maxi = dp[start][end].second;
            return dp[start][end].first;
        }
        int ret = INT_MAX;
        
        for(int i = start;i < end; ++i){
            int maxL=0;
            int a = f(nList,start,i,maxL);
            int maxR = 0;
            int b = f(nList,i+1,end,maxR);
            ret = min(ret,maxL* maxR + a + b);
            maxi = max(maxi,nList[i]);
        }
        maxi = max(maxi,nList[end]);
        dp[start][end] = {ret,maxi};
        
        return ret;
    }
    int mctFromLeafValues(vector<int>& nList) {
        int N = nList.size();
        dp.resize(N,vector<pair<int,int>>(N,{-1,-1}));
        int tmp = 0;
        return f(nList,0,N-1,tmp);
    }
};
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