<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 node {
    public: 
        int sum;
        int maxNum;
        int minNum;
        node* left;
        node* right;
        bool isBst;

        node() {
            this->sum =0;
            this->maxNum = INT_MIN;
            this->minNum = INT_MAX;
            this->isBst = true;
            this->left = NULL;
            this->right = NULL;
        }
};

class Solution {
public:
    node* createNode(TreeNode* root) {
        if (root == NULL) {
            return NULL;
        }
        node* newNode = new node();
        newNode->sum = root->val;
        bool isBst = true;

        if (root->left) {
            newNode->left = createNode(root->left);
            if (!newNode->left->isBst || newNode->left->maxNum >= root->val) {
                isBst = false;
            }
            newNode->sum += newNode->left->sum;
            newNode->minNum = min(newNode->minNum, newNode->left->minNum);
            newNode->maxNum = max(newNode->maxNum, newNode->left->maxNum);
        }

        if (root->right) {
            newNode->right = createNode(root->right);
            if (!newNode->right->isBst || newNode->right->minNum <= root->val) {
                isBst = false;
            }
            newNode->sum += newNode->right->sum;
            newNode->minNum = min(newNode->minNum, newNode->right->minNum);
            newNode->maxNum = max(newNode->maxNum, newNode->right->maxNum);
        }

        newNode->isBst = isBst;
        newNode->minNum = min(newNode->minNum, root->val);
        newNode->maxNum = max(newNode->maxNum, root->val);

        return newNode;
    }
    void findAns(node* root, int &ans) {
        if (root == NULL) {
            return;
        }
        if (root->isBst) {
            ans = max(ans, root->sum);
        }
        findAns(root->left, ans);
        findAns(root->right, ans);
    }
    int maxSumBST(TreeNode* root) {
        node* rote = createNode(root);
        int ans = 0;
        findAns(rote, ans);
        return ans;
    }
};
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