<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) {}
 * };
 */
class Solution {
public:
    vector<int>ms(TreeNode* root,int& ans){
        if(!root->left && !root->right){
            ans=max(ans,root->val);
            return {root->val,root->val,root->val};
        }
        vector<int>v,v1;
        if(!root->right){
            v=ms(root->left,ans);
            // cout<<v[0]<<v[1]<<v[2]<<endl;
            if(v[0]!=INT_MIN && v[2]<root->val){
                ans=max(ans,v[0]+root->val);
                // cout<<v[0]+root->val<<v[1]<<root->val<<endl;
                return {v[0]+root->val,v[1],root->val};
            }
            return {INT_MIN,-1,-1};
        }
        if(!root->left){
            v=ms(root->right,ans);
            // cout<<v[0]<<v[1]<<v[2]<<endl;
            if(v[0]!=INT_MIN && v[1]>root->val){
                ans=max(ans,v[0]+root->val);
                // cout<<v[0]+root->val<<root->val<<v[2]<<endl;
                return {v[0]+root->val,root->val,v[2]};
            }
            return {INT_MIN,-1,-1};
        }
        v=ms(root->left,ans);
        // cout<<v[0]<<v[1]<<v[2]<<endl;
        v1=ms(root->right,ans);
        // cout<<v1[0]<<v1[1]<<v1[2]<<endl;
        if(v[0]!=INT_MIN && v1[0]!=INT_MIN && v[2]<root->val && v1[1]>root->val){
            ans=max(ans,v[0]+v1[0]+root->val);
            // cout<<
            return {v[0]+v1[0]+root->val,v[1],v1[2]};
        }
        return {INT_MIN,-1,-1};
    }
    int maxSumBST(TreeNode* root) {
        int ans=0;
        ms(root,ans);
        return ans;
    }
};
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