<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:
    int ans = 0;
    vector<int> dfs(TreeNode* root){ // min , max , sum , isValid
        vector<int> itl , itr;
        if(!root) return{INT_MIN , INT_MAX , 0 ,1}; 
        if(!root->left && !root->right){
            ans = max(ans , root->val);
            return {root->val , root->val , root->val , 1};
        } 

        else if(!root->left){
            itr = dfs(root->right);
            if(itr[0] > root->val && itr[3]){
                ans = max(ans , itr[2] + root->val);
                return {root->val , itr[1] , itr[2] + root->val , 1};
            }
            return {root->val , itr[1] , itr[2] + root->val , 0};
        }  

        else if(!root->right){
            itl = dfs(root->left);
            if(itl[1] < root->val && itl[3]){
                ans = max(ans , itl[2] + root->val);
                return {itl[0] , root->val , itl[2] + root->val , 1};
            }
            return {itl[0] , root->val , itl[2] + root->val , 0};
        } 

        else {
            auto itl = dfs(root->left) , itr = dfs(root->right);

        if(itl[1] < root->val && itr[0] > root->val && itl[3] && itr[3]){
            cout << root->val << " " << itl[0] << " " << itr[1] << endl;
            ans = max(ans , itl[2] + itr[2] + root->val);
            return {itl[0] , itr[1] , itl[2] + itr[2] + root->val , 1};
        }
        return {itl[0] , itr[1] , itl[2] + itr[2] + root->val , 0}; 
        }

    }
    int maxSumBST(TreeNode* root) {
        dfs(root);       
        return ans; 
    }
};
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