<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 solveUsingRec(vector<int>& arr, int start, int end, map< pair<int, int>, int>& mp)
    {
        if(start >= end)
        {
            return 0;
        }
        int ans = INT_MAX;
        for(int i = start; i < end; i++)
        {
            ans = min(ans, (mp[{start, i}] * mp[{i + 1, end}]) + solveUsingRec(arr, start, i,mp) + 
            solveUsingRec(arr, i + 1, end, mp));
        }
        return ans;
    }

    int solveUsingMem(vector<int>& arr, int start, int end, map< pair<int, int>, int>& mp, vector<vector<int>>& dp)
    {
        if(start >= end)
        {
            return 0;
        }
        if(dp[start][end] != -1)
        {
            return dp[start][end];
        }
        int ans = INT_MAX;
        for(int i = start; i < end; i++)
        {
            ans = min(ans, (mp[{start, i}] * mp[{i + 1, end}]) + solveUsingMem(arr, start, i,mp, dp) + 
            solveUsingMem(arr, i + 1, end, mp, dp));
        }
        dp[start][end] = ans;
        return dp[start][end];
    }
    int mctFromLeafValues(vector<int>& arr) {
        map< pair<int, int> , int> mp;
        for(int i = 0; i < arr.size();i++)
        {
            mp[{i,i}] = arr[i];
            for(int j = i + 1; j < arr.size();j++)
            {
                mp[{i, j}] = max(arr[j], mp[{i, j -1}]);
            }
        }
        int n = arr.size();
        // return solveUsingRec(arr, 0, n -1, mp);
        vector<vector<int>>dp(n + 1, vector<int>(n + 1, -1));
        return solveUsingMem(arr, 0, n-1, mp, dp);
    }
};
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