<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;
    class BST{
        public:
        int mini;
        int maxi;
        bool isbst;
        int sum;

        BST(int a,int b,int c,bool d){
            mini = a;
            maxi = b;
            sum = c;
            isbst = d;
        }
    };

    BST* help(TreeNode* root){
        if(root==nullptr){
            BST* temp = new BST(INT_MAX,INT_MIN,0,true);
            return temp;
        }
        cout<<" for root "<<root->val<<endl;

        if(root->left == nullptr && root->right == nullptr){
            ans = max(ans,root->val);
            BST* temp = new BST(root->val,root->val,root->val,true);
            return temp;
        }
        if(root->left == nullptr){
            BST* right = help(root->right);
            bool fl = false;
            int mini = right ->mini;
            if(right->isbst){
                if( root->val>=mini)fl= false;
                else fl = true;
            }
            int sum = root->val + right->sum;
            mini = min(root->val,mini);
            int maxi = max(root->val,right->maxi);

            if (fl){
                cout<<" sum "<<sum<<endl;
                ans = max(sum,ans);
            }
            BST* temp = new BST(mini,maxi,sum,fl);
            return temp;

        }

        if(root->right == nullptr){
            BST* left = help(root->left);
            bool fl = false;
            int maxi = left ->maxi;
            if(left->isbst){
                if( root->val<=maxi)fl= false;
                else fl = true;
            }
            int sum = left->sum + root->val;
            int mini = min(root->val,left->mini);
            maxi = max(root->val,maxi);
            if (fl){
                cout<<" sum "<<sum<<endl;
                ans = max(sum,ans);
            }

            BST* temp = new BST(mini,maxi,sum,fl);
            return temp;
        }



        BST* left = help(root->left);
        BST* right = help(root->right);

        int sum = left->sum + root->val + right->sum;
        cout<<" is left bst "<<left->isbst<<"  and is right bst "<<right->isbst<<endl;

        int maxi =left->maxi ;    //max(max(root->val,left->maxi),right->maxi);
        int mini = right->mini;    //min(min(root->val,left->min),right->min);

        bool fl = false;
        if(left->isbst && right->isbst){
            if(root->val<=maxi || root->val>=mini)fl= false;
            else fl = true;
        }
        cout<<" is root bst "<<fl<<endl;

        if (fl){
            cout<<" sum "<<sum<<endl;
            ans = max(sum,ans);
        }

        maxi = max(max(root->val,left->maxi),right->maxi);
        mini = min(min(root->val,left->mini),right->mini);

        BST* temp = new BST(mini,maxi,sum,fl);
        return temp;
    }


    int maxSumBST(TreeNode* root) {
        BST* T = help(root);
        return ans;
        
    }
};
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