<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 solveUsingRecursion(vector<int>& arr, map< pair<int,int>, int >& maxi, int s, int e) {
        //base case
        if(s > e) {
            return 0;
        }
        if(s == e) {
            return 0;
        } 
        int ans = INT_MAX;
        for(int i=s; i<e; i++) {
            //i is used here for partitioning
            ans = min(ans, (maxi[{s, i}] * maxi[{i+1,e}]) + 
                            solveUsingRecursion(arr,maxi, s, i) + 
                            solveUsingRecursion(arr,maxi, i+1, e) );
        }
        return ans;
    }

    int solveUsingMem(vector<int>& arr, map< pair<int,int>, int >& maxi, int s, int e,vector<vector<int> >& dp) {
        //base case
        if(s > e) {
            return 0;
        }
        if(s == e) {
            return 0;
        } 
        if(dp[s][e] != -1) {
            return dp[s][e];
        }
        int ans = INT_MAX;
        for(int i=s; i<e; i++) {
            //i is used here for partitioning
            ans = min(ans, (maxi[{s, i}] * maxi[{i+1,e}]) + 
                            solveUsingMem(arr,maxi, s, i,dp) + 
                            solveUsingMem(arr,maxi, i+1, e, dp) );
        }
        dp[s][e] = ans;
        return dp[s][e];
    }

    int solveUsingTabulation(vector<int>& arr,map< pair<int,int>, int >& maxi) {
        int n = arr.size();
        vector<vector<int> > dp(n+2, vector<int>(n+1, 0));

        for(int s_index=n; s_index>=0; s_index--) {
            for(int e_index=0; e_index<=n-1; e_index++) {
                if(s_index >= e_index) {
                    continue;
                }
                int ans = INT_MAX;
                for(int i=s_index; i<e_index; i++) {
                    //i is used here for partitioning
                    ans = min(ans, (maxi[{s_index, i}] * maxi[{i+1,e_index}]) + dp[s_index][i] +  dp[i+1][e_index] );
                }
                dp[s_index][e_index] = ans;
            }
        }
         return dp[0][n-1];
    }

    int mctFromLeafValues(vector<int>& arr) {
         //pre computation
        map< pair<int,int>, int > maxi;
        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}]);
            }
        }
        int n = arr.size();
        int start = 0;
        int end = n-1;
        vector<vector<int> > dp(n+1, vector<int>(n+1, -1));
        int ans = solveUsingTabulation(arr,maxi);
        return ans;
    }
};
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