<p>Given a <strong>binary tree</strong> <code>root</code>, return <em>the maximum sum of all keys of <strong>any</strong> sub-tree which is also a Binary Search Tree (BST)</em>.</p>

<p>Assume a BST is defined as follows:</p>

<ul>
	<li>The left subtree of a node contains only nodes with keys <strong>less than</strong> the node&#39;s key.</li>
	<li>The right subtree of a node contains only nodes with keys <strong>greater than</strong> the node&#39;s key.</li>
	<li>Both the left and right subtrees must also be binary search trees.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/01/30/sample_1_1709.png" style="width: 320px; height: 250px;" /></p>

<pre>
<strong>Input:</strong> root = [1,4,3,2,4,2,5,null,null,null,null,null,null,4,6]
<strong>Output:</strong> 20
<strong>Explanation:</strong> Maximum sum in a valid Binary search tree is obtained in root node with key equal to 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/01/30/sample_2_1709.png" style="width: 134px; height: 180px;" /></p>

<pre>
<strong>Input:</strong> root = [4,3,null,1,2]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Maximum sum in a valid Binary search tree is obtained in a single root node with key equal to 2.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> root = [-4,-2,-5]
<strong>Output:</strong> 0
<strong>Explanation:</strong> All values are negatives. Return an empty BST.
</pre>

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

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

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */

 struct Node {
     int val;
     int maxLeft=-1e8;
     int minRight = 1e8;
     Node *left;
     Node *right;
     Node() : val(0), left(nullptr), right(nullptr) {}
     Node(int x) : val(x), left(nullptr), right(nullptr) {}
     Node(int x, Node *left, Node *right) : val(x), left(left),right(right) {}
  };

class Solution {
public:
    int ans = 0;

    int maxLeft(TreeNode* root,Node* newRoot){
        if(!root){
            return -1e8;
        }
        newRoot->val = root->val;
        if(root->left){
            Node* newLeft = new Node();
            newRoot->left = newLeft;
        }
        if(root->right){
            Node* newRight = new Node();
            newRoot->right = newRight;
        }

        int lt = maxLeft(root->left,newRoot->left);
        int rt = maxLeft(root->right,newRoot->right);
        newRoot->maxLeft = lt;
        return max(root->val,max(lt,rt));
    }
    int minRight(TreeNode* root,Node* node){
        if(!root){
            return 1e8;
        }

        int rt = minRight(root->right,node->right);
        int lt = minRight(root->left,node->left);

        node->minRight = rt;

        return min(root->val,min(lt,rt));
    }

    int helper(TreeNode* root,Node* node){
        if(!root){
            return 0;
        }
          int left = helper(root->left,node->left);
          int right = helper(root->right,node->right);
            if(right == -1e8 || left == -1e8){
                return right;
            }
          if(root->val > node->maxLeft && root->val < node->minRight){
            ans = max(ans,left+right+root->val);
            return left+right+root->val;
          } else {
            return -1e8;
          }    
    }

    int maxSumBST(TreeNode* root) {

        Node* node = new Node();

        maxLeft(root,node);
        minRight(root,node);        
        helper(root,node);

        return ans;
    }
};
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