<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 box{
    public:
     bool bst;
     int sum;
     int max;
     int min;
     box(int data) {
        bst = 1;
        sum = data;
        max = data;
        min = data;
     }
 };
class Solution {
public:
    box* find(TreeNode * root, int & sum) {
        box * t = new box(0);
        if(root == NULL ) {
            t->bst = 1;
            t->sum = 0;
            t->min = INT_MAX;
            t->max = INT_MIN;
            return t;
        }
        else if(root->left == NULL && root->right == NULL) {
            t->bst = 1;
            t->sum = root->val;
            sum = max(sum,t->sum);
            t->min = root->val;
            t->max = root->val;
            return t;

        }

        box* left = find(root->left, sum);
        box * right = find(root->right, sum);

        if(left->bst && right ->bst) {
            if(root->val > left->max && root->val< right->min) {
                t->bst = 1;
                t->sum = root->val + left->sum + right->sum;
                sum = max(t->sum, sum);
                t->min = min(root->val, min(left->min, right->min));
                t->max = max(root->val, max(left->max, right->max));
                return t;
            }
            else {
                t->bst = 0;
                t->sum = root->val + left->sum + right->sum;
                // sum = max(t->sum, sum);
                t->min = min(root->val, min(left->min, right->min));
                t->max = max(root->val, max(left->max, right->max));
                return t;
            }
        }
                t->bst = 0;
                t->sum = root->val + left->sum + right->sum;
                // sum = max(t->sum, sum);
                t->min = min(root->val, min(left->min, right->min));
                t->max = max(root->val, max(left->max, right->max));
                return t;
        
    }
    int maxSumBST(TreeNode* root) {
        int sum = 0;
        box * t = find(root, sum);
        return sum;
    }
};
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