<p>Given the <code>root</code> of a binary tree, split the binary tree into two subtrees by removing one edge such that the product of the sums of the subtrees is maximized.</p>

<p>Return <em>the maximum product of the sums of the two subtrees</em>. Since the answer may be too large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p><strong>Note</strong> that you need to maximize the answer before taking the mod and not after taking it.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/01/21/sample_1_1699.png" style="width: 500px; height: 167px;" />
<pre>
<strong>Input:</strong> root = [1,2,3,4,5,6]
<strong>Output:</strong> 110
<strong>Explanation:</strong> Remove the red edge and get 2 binary trees with sum 11 and 10. Their product is 110 (11*10)
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/01/21/sample_2_1699.png" style="width: 500px; height: 211px;" />
<pre>
<strong>Input:</strong> root = [1,null,2,3,4,null,null,5,6]
<strong>Output:</strong> 90
<strong>Explanation:</strong> Remove the red edge and get 2 binary trees with sum 15 and 6.Their product is 90 (15*6)
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the tree is in the range <code>[2, 5 * 10<sup>4</sup>]</code>.</li>
	<li><code>1 &lt;= Node.val &lt;= 10<sup>4</sup></code></li>
</ul>

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right

"""

intution in one pass calc all the sum of tree, in second pass calculate lsum and right sum
calculate the product and maximize the ans

TC : O(2N)
SC : O(N)

"""

class Solution:
    def maxProduct(self, root: Optional[TreeNode]) -> int:

        total_sum = self.calc_sum(root)
        max_prod = [0]

        self.max_prod(root, max_prod, total_sum)

        MOD = (10 ** 9) + 7

        return max_prod[0] % (MOD)
    
    def max_prod(self, root: Optional[TreeNode], max_prod: List[int], total_sum: int) -> int:

        if not root:
            return 0

        l_sum = self.max_prod(root.left, max_prod, total_sum)
        r_sum = self.max_prod(root.right, max_prod, total_sum)

        max_prod[0] = max(max_prod[0], (total_sum - l_sum) * l_sum)
        max_prod[0] = max(max_prod[0], (total_sum - r_sum) * r_sum)



        return l_sum + r_sum + root.val
    
    def calc_sum(self, root: Optional[TreeNode]) -> int:

        if not root:
            return 0
        
        l = self.calc_sum(root.left)
        r = self.calc_sum(root.right)

        return root.val + l + r

        
        
        
202