<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 {
    private:
    int solve(vector<int>& arr,  map<pair<int,int>,int>& maxi,int str,int end){
        if(str==end){
            return 0;
        }
        int ans=INT_MAX;

        for(int i=str;i<end;i++){
            ans=min(ans,maxi[{str,i}]*maxi[{i+1,end}]+solve(arr,maxi,str,i)+solve(arr,maxi,i+1,end));

        }
        return ans;
    }

    int solveMem(vector<int>& arr,  map<pair<int,int>,int>& maxi,int str,int end, vector<vector<int>>& dp){
        if(str==end){
            return 0;
        }
        if(dp[str][end]!= -1){
            return dp[str][end];
        }
        int ans=INT_MAX;

        for(int i=str;i<end;i++){
            ans=min(ans,maxi[{str,i}]*maxi[{i+1,end}]+solveMem(arr,maxi,str,i,dp)+solveMem(arr,maxi,i+1,end,dp));

        }
        return dp[str][end]=ans;
    }
    int solveTab(vector<int>& arr,  map<pair<int,int>,int>& maxi,int n){
       vector<vector<int>> dp(n+1,vector<int>(n+1,0)); 


       for(int str=n-1;str>=0;str--){
        for(int end=str+1;end<n;end++){
             int ans=INT_MAX;

        for(int i=str;i<end;i++){
            ans=min(ans,maxi[{str,i}]*maxi[{i+1,end}]+dp[str][i]+dp[i+1][end]);

        }
         dp[str][end]=ans;
        }
        
       }
      return dp[0][n-1];


    }

public:
    int mctFromLeafValues(vector<int>& arr) {
       map<pair<int,int>,int> maxi;
        int n=arr.size();

        for(int i=0;i<arr.size();i++){
            maxi[{i,i}]=arr[i];
            for(int j=i+1;j<arr.size();j++){
                maxi[{i,j}]=max(arr[j],maxi[{i,j-1}]);
            }
        }
        vector<vector<int>> dp(n+1,vector<int>(n+1,-1));
        return solveTab(arr,maxi,n);
    }
};
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