# Maximum sub-tree sum in a Binary Tree such that the sub-tree is also a BST

Given a binary tree, the task is to print the maximum sum of nodes of a sub-tree which is also a Binary Search Tree.
Examples:

```Input :
7
/  \
12    2
/  \    \
11  13    5
/         / \
2         1   38

Output:44
BST rooted under node 5 has the maximum sum
5
/ \
1   38

Input:
5
/  \
9    2
/      \
6        3
/ \
8   7

Output: 8
Here each leaf node represents a binary search tree
also a BST with sum 5 exists
2
\
3
But the leaf node 8 has the maximum sum.
```

Approach: We traverse the tree in bottom-up manner. For every traversed node, we store the information of maximum and minimum of that subtree, a variable isBST to store if it is a BST, variable currmax to store the maximum sum of BST found till now, and a variable sum to store the sum of Left and Right subtree(which is also a BST) rooted under the current node.
Below is the implementation of the above approach:

 `// C++ implementation of the approach` `#include ` `using` `namespace` `std;`   `// Binary tree node` `struct` `Node {` `    ``struct` `Node* left;` `    ``struct` `Node* right;` `    ``int` `data;`   `    ``Node(``int` `data)` `    ``{` `        ``this``->data = data;` `        ``this``->left = NULL;` `        ``this``->right = NULL;` `    ``}` `};`   `// Information stored in every` `// node during bottom up traversal` `struct` `Info {`   `    ``// Max Value in the subtree` `    ``int` `max;`   `    ``// Min value in the subtree` `    ``int` `min;`   `    ``// If subtree is BST` `    ``bool` `isBST;`   `    ``// Sum of the nodes of the sub-tree` `    ``// rooted under the current node` `    ``int` `sum;`   `    ``// Max sum of BST found till now` `    ``int` `currmax;` `};`   `// Returns information about subtree such as` `// subtree with maximum sum which is also a BST` `Info MaxSumBSTUtil(``struct` `Node* root, ``int``& maxsum)` `{` `    ``// Base case` `    ``if` `(root == NULL)` `        ``return` `{ INT_MIN, INT_MAX, ``true``, 0, 0 };`   `    ``// If current node is a leaf node then` `    ``// return from the function and store` `    ``// information about the leaf node` `    ``if` `(root->left == NULL && root->right == NULL) {` `        ``maxsum = max(maxsum, root->data);` `        ``return` `{ root->data, root->data, ``true``, root->data, maxsum };` `    ``}`   `    ``// Store information about the left subtree` `    ``Info L = MaxSumBSTUtil(root->left, maxsum);`   `    ``// Store information about the right subtree` `    ``Info R = MaxSumBSTUtil(root->right, maxsum);`   `    ``Info BST;`   `    ``// If the subtree rooted under the current node` `    ``// is a BST` `    ``if` `(L.isBST && R.isBST && L.max < root->data && R.min > root->data) {`   `        ``BST.max = max(root->data, max(L.max, R.max));` `        ``BST.min = min(root->data, min(L.min, R.min));`   `        ``maxsum = max(maxsum, R.sum + root->data + L.sum);` `        ``BST.sum = R.sum + root->data + L.sum;`   `        ``// Update the current maximum sum` `        ``BST.currmax = maxsum;`   `        ``BST.isBST = ``true``;` `        ``return` `BST;` `    ``}`   `    ``// If the whole tree is not a BST then` `    ``// update the current maximum sum` `    ``BST.isBST = ``false``;` `    ``BST.currmax = maxsum;` `    ``BST.sum = R.sum + root->data + L.sum;`   `    ``return` `BST;` `}`   `// Function to return the maximum` `// sum subtree which is also a BST` `int` `MaxSumBST(``struct` `Node* root)` `{` `    ``int` `maxsum = INT_MIN;` `    ``return` `MaxSumBSTUtil(root, maxsum).currmax;` `}`   `// Driver code` `int` `main()` `{` `    ``struct` `Node* root = ``new` `Node(5);` `    ``root->left = ``new` `Node(14);` `    ``root->right = ``new` `Node(3);` `    ``root->left->left = ``new` `Node(6);` `    ``root->right->right = ``new` `Node(7);` `    ``root->left->left->left = ``new` `Node(9);` `    ``root->left->left->right = ``new` `Node(1);`   `    ``cout << MaxSumBST(root);`   `    ``return` `0;` `}`

 `// Java implementation of the approach` `class` `GFG` `{` `    `  `// Binary tree node` `static` `class` `Node ` `{` `    ``Node left;` `    ``Node right;` `    ``int` `data;`   `    ``Node(``int` `data)` `    ``{` `        ``this``.data = data;` `        ``this``.left = ``null``;` `        ``this``.right = ``null``;` `    ``}` `};`   `// Information stored in every` `// node during bottom up traversal` `static` `class` `Info ` `{`   `    ``// Max Value in the subtree` `    ``int` `max;`   `    ``// Min value in the subtree` `    ``int` `min;`   `    ``// If subtree is BST` `    ``boolean` `isBST;`   `    ``// Sum of the nodes of the sub-tree` `    ``// rooted under the current node` `    ``int` `sum;`   `    ``// Max sum of BST found till now` `    ``int` `currmax;` `    `  `    ``Info(``int` `m,``int` `mi,``boolean` `is,``int` `su,``int` `cur)` `    ``{` `        ``max = m;` `        ``min = mi;` `        ``isBST = is;` `        ``sum = su;` `        ``currmax = cur;` `    ``}` `    ``Info(){}` `};`   `static` `class` `INT` `{` `    ``int` `a;` `}`   `// Returns information about subtree such as` `// subtree with the maximum sum which is also a BST` `static` `Info MaxSumBSTUtil( Node root, INT maxsum)` `{` `    ``// Base case` `    ``if` `(root == ``null``)` `        ``return` `new` `Info( Integer.MIN_VALUE, ` `                        ``Integer.MAX_VALUE, ``true``, ``0``, ``0` `);`   `    ``// If current node is a leaf node then` `    ``// return from the function and store` `    ``// information about the leaf node` `    ``if` `(root.left == ``null` `&& root.right == ``null``)` `    ``{` `        ``maxsum.a = Math.max(maxsum.a, root.data);` `        ``return` `new` `Info( root.data, root.data, ` `                        ``true``, root.data, maxsum.a );` `    ``}`   `    ``// Store information about the left subtree` `    ``Info L = MaxSumBSTUtil(root.left, maxsum);`   `    ``// Store information about the right subtree` `    ``Info R = MaxSumBSTUtil(root.right, maxsum);`   `    ``Info BST=``new` `Info();`   `    ``// If the subtree rooted under the current node` `    ``// is a BST` `    ``if` `(L.isBST && R.isBST && L.max < root.data && ` `                               ``R.min > root.data)` `    ``{`   `        ``BST.max = Math.max(root.data, Math.max(L.max, R.max));` `        ``BST.min = Math.min(root.data, Math.min(L.min, R.min));`   `        ``maxsum.a = Math.max(maxsum.a, R.sum + root.data + L.sum);` `        ``BST.sum = R.sum + root.data + L.sum;`   `        ``// Update the current maximum sum` `        ``BST.currmax = maxsum.a;`   `        ``BST.isBST = ``true``;` `        ``return` `BST;` `    ``}`   `    ``// If the whole tree is not a BST then` `    ``// update the current maximum sum` `    ``BST.isBST = ``false``;` `    ``BST.currmax = maxsum.a;` `    ``BST.sum = R.sum + root.data + L.sum;`   `    ``return` `BST;` `}`   `// Function to return the maximum` `// sum subtree which is also a BST` `static` `int` `MaxSumBST( Node root)` `{` `    ``INT maxsum = ``new` `INT();` `    ``maxsum.a = Integer.MIN_VALUE;` `    ``return` `MaxSumBSTUtil(root, maxsum).currmax;` `}`   `// Driver code` `public` `static` `void` `main(String args[])` `{` `    ``Node root = ``new` `Node(``5``);` `    ``root.left = ``new` `Node(``14``);` `    ``root.right = ``new` `Node(``3``);` `    ``root.left.left = ``new` `Node(``6``);` `    ``root.right.right = ``new` `Node(``7``);` `    ``root.left.left.left = ``new` `Node(``9``);` `    ``root.left.left.right = ``new` `Node(``1``);`   `    ``System.out.println( MaxSumBST(root));` `}` `}`   `// This code is contributed by Arnab Kundu`

 `# Python3 implementation of ` `# the above approach` `from` `sys ``import` `maxsize as INT_MAX` `INT_MIN ``=` `-``INT_MAX`   `# Binary tree node` `class` `Node:` `  `  `    ``def` `__init__(``self``, data):` `        ``self``.data ``=` `data` `        ``self``.left ``=` `None` `        ``self``.right ``=` `None`   `# Information stored in every` `# node during bottom up traversal` `class` `Info:` `  `  `    ``def` `__init__(``self``, _max, _min, ` `                 ``isBST, _sum, currmax):` `      `  `        ``# Max Value in the subtree` `        ``self``.``max` `=` `_max`   `        ``# Min value in the subtree` `        ``self``.``min` `=` `_min`   `        ``# If subtree is BST` `        ``self``.isBST ``=` `isBST`   `        ``# Sum of the nodes of the sub-tree` `        ``# rooted under the current node` `        ``self``.``sum` `=` `_sum`   `        ``# Max sum of BST found till now` `        ``self``.currmax ``=` `currmax`   `# Returns information about ` `# subtree such as subtree ` `# with maximum sum which ` `# is also a BST` `def` `MaxSumBSTUtil(root: Node) ``-``> Info:` `    ``global` `maxsum`   `    ``# Base case` `    ``if` `(root ``is` `None``):` `        ``return` `Info(INT_MIN, INT_MAX, ` `                    ``True``, ``0``, ``0``)`   `    ``# If current node is a ` `    ``# leaf node then return ` `    ``# from the function and store` `    ``# information about the leaf node` `    ``if` `(root.left ``is` `None` `and` `        ``root.right ``is` `None``):` `        ``maxsum ``=` `max``(maxsum, ` `                     ``root.data)` `        ``return` `Info(root.data, root.data, ` `                    ``True``, root.data, maxsum)`   `    ``# Store information about ` `    ``# the left subtree` `    ``L ``=` `MaxSumBSTUtil(root.left)`   `    ``# Store information about ` `    ``# the right subtree` `    ``R ``=` `MaxSumBSTUtil(root.right)`   `    ``BST ``=` `Info`   `    ``# If the subtree rooted under ` `    ``# the current node is a BST` `    ``if` `(L.isBST ``and` `R.isBST ``and` `        ``L.``max` `< root.data ``and` `        ``R.``min` `> root.data):`   `        ``BST.``max` `=` `max``(root.data, ` `                  ``max``(L.``max``, R.``max``))` `        ``BST.``min` `=` `min``(root.data, ` `                  ``min``(L.``min``, R.``min``))`   `        ``maxsum ``=` `max``(maxsum, R.``sum` `+` `                     ``root.data ``+` `L.``sum``)` `        ``BST.``sum` `=` `R.``sum` `+` `root.data ``+` `L.``sum`   `        ``# Update the current maximum sum` `        ``BST.currmax ``=` `maxsum`   `        ``BST.isBST ``=` `True` `        ``return` `BST`   `    ``# If the whole tree is not ` `    ``# a BST then update the ` `    ``# current maximum sum` `    ``BST.isBST ``=` `False` `    ``BST.currmax ``=` `maxsum` `    ``BST.``sum` `=` `R.``sum` `+` `root.data ``+` `L.``sum`   `    ``return` `BST`   `# Function to return the maximum` `# sum subtree which is also a BST` `def` `MaxSumBST(root: Node) ``-``> ``int``:` `    ``global` `maxsum` `    ``return` `MaxSumBSTUtil(root).currmax`   `# Driver code` `if` `__name__ ``=``=` `"__main__"``:`   `    ``root ``=` `Node(``5``)` `    ``root.left ``=` `Node(``14``)` `    ``root.right ``=` `Node(``3``)` `    ``root.left.left ``=` `Node(``6``)` `    ``root.right.right ``=` `Node(``7``)` `    ``root.left.left.left ``=` `Node(``9``)` `    ``root.left.left.right ``=` `Node(``1``)`   `    ``maxsum ``=` `INT_MIN` `    ``print``(MaxSumBST(root))`   `# This code is contributed by sanjeev2552`

 `// C# implementation of the approach` `using` `System;`   `class` `GFG` `{` `    `  `// Binary tree node` `public` `class` `Node ` `{` `    ``public` `Node left;` `    ``public` `Node right;` `    ``public` `int` `data;`   `    ``public` `Node(``int` `data)` `    ``{` `        ``this``.data = data;` `        ``this``.left = ``null``;` `        ``this``.right = ``null``;` `    ``}` `};`   `// Information stored in every` `// node during bottom up traversal` `public` `class` `Info ` `{`   `    ``// Max Value in the subtree` `    ``public` `int` `max;`   `    ``// Min value in the subtree` `    ``public` `int` `min;`   `    ``// If subtree is BST` `    ``public` `bool` `isBST;`   `    ``// Sum of the nodes of the sub-tree` `    ``// rooted under the current node` `    ``public` `int` `sum;`   `    ``// Max sum of BST found till now` `    ``public` `int` `currmax;` `    `  `    ``public` `Info(``int` `m,``int` `mi,``bool` `s,``int` `su,``int` `cur)` `    ``{` `        ``max = m;` `        ``min = mi;` `        ``isBST = s;` `        ``sum = su;` `        ``currmax = cur;` `    ``}` `    ``public` `Info(){}` `};`   `public` `class` `INT` `{` `    ``public` `int` `a;` `}`   `// Returns information about subtree such as` `// subtree with the maximum sum which is also a BST` `static` `Info MaxSumBSTUtil( Node root, INT maxsum)` `{` `    ``// Base case` `    ``if` `(root == ``null``)` `        ``return` `new` `Info( ``int``.MinValue, ` `                        ``int``.MaxValue, ``true``, 0, 0 );`   `    ``// If current node is a leaf node then` `    ``// return from the function and store` `    ``// information about the leaf node` `    ``if` `(root.left == ``null` `&& root.right == ``null``)` `    ``{` `        ``maxsum.a = Math.Max(maxsum.a, root.data);` `        ``return` `new` `Info( root.data, root.data, ` `                        ``true``, root.data, maxsum.a );` `    ``}`   `    ``// Store information about the left subtree` `    ``Info L = MaxSumBSTUtil(root.left, maxsum);`   `    ``// Store information about the right subtree` `    ``Info R = MaxSumBSTUtil(root.right, maxsum);`   `    ``Info BST = ``new` `Info();`   `    ``// If the subtree rooted under the current node` `    ``// is a BST` `    ``if` `(L.isBST && R.isBST && L.max < root.data && ` `                            ``R.min > root.data)` `    ``{`   `        ``BST.max = Math.Max(root.data, Math.Max(L.max, R.max));` `        ``BST.min = Math.Min(root.data, Math.Min(L.min, R.min));`   `        ``maxsum.a = Math.Max(maxsum.a, R.sum + root.data + L.sum);` `        ``BST.sum = R.sum + root.data + L.sum;`   `        ``// Update the current maximum sum` `        ``BST.currmax = maxsum.a;`   `        ``BST.isBST = ``true``;` `        ``return` `BST;` `    ``}`   `    ``// If the whole tree is not a BST then` `    ``// update the current maximum sum` `    ``BST.isBST = ``false``;` `    ``BST.currmax = maxsum.a;` `    ``BST.sum = R.sum + root.data + L.sum;`   `    ``return` `BST;` `}`   `// Function to return the maximum` `// sum subtree which is also a BST` `static` `int` `MaxSumBST( Node root)` `{` `    ``INT maxsum = ``new` `INT();` `    ``maxsum.a = ``int``.MinValue;` `    ``return` `MaxSumBSTUtil(root, maxsum).currmax;` `}`   `// Driver code` `public` `static` `void` `Main(String []args)` `{` `    ``Node root = ``new` `Node(5);` `    ``root.left = ``new` `Node(14);` `    ``root.right = ``new` `Node(3);` `    ``root.left.left = ``new` `Node(6);` `    ``root.right.right = ``new` `Node(7);` `    ``root.left.left.left = ``new` `Node(9);` `    ``root.left.left.right = ``new` `Node(1);`   `    ``Console.WriteLine( MaxSumBST(root));` `}` `}`   `// This code has been contributed by 29AjayKumar`

Output:
```10

```

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