Largest BST in a Binary Tree | Set 2

• Difficulty Level : Hard
• Last Updated : 23 Dec, 2021

Given a Binary Tree, write a function that returns the size of the largest subtree which is also a Binary Search Tree (BST). If the complete Binary Tree is BST, then return the size of the whole tree.
Examples:

Input:
5
/  \
2    4
/  \
1    3

Output: 3
The following subtree is the
maximum size BST subtree
2
/  \
1    3

Input:
50
/    \
30       60
/  \     /  \
5   20   45    70
/  \
65    80
Output: 5
The following subtree is the
maximum size BST subtree
60
/  \
45    70
/  \
65    80

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

We have discussed two methods in below post.
Find the largest BST subtree in a given Binary Tree | Set 1
In this post, a different O(n) solution is discussed. This solution is simpler than the solutions discussed above and works in O(n) time.
The idea is based on method 3 of check if a binary tree is BST article
A Tree is BST if following is true for every node x.

1. The largest value in left subtree (of x) is smaller than value of x.
2. The smallest value in right subtree (of x) is greater than value of x.

We traverse tree in bottom up manner. For every traversed node, we return maximum and minimum values in subtree rooted with it. If any node follows above properties and size of

C++

 // C++ program to find largest BST in a// Binary Tree.#include using namespace std; /* A binary tree node has data,pointer to left child and apointer to right child */struct Node{    int data;    struct Node* left;    struct Node* right;}; /* Helper function that allocates a newnode with the given data and NULL leftand right pointers. */struct Node* newNode(int data){    struct Node* node = new Node;    node->data = data;    node->left = node->right = NULL;     return(node);} // Information to be returned by every// node in bottom up traversal.struct Info{    int sz; // Size of subtree    int max; // Min value in subtree    int min; // Max value in subtree    int ans; // Size of largest BST which    // is subtree of current node    bool isBST; // If subtree is BST}; // Returns Information about subtree. The// Information also includes size of largest// subtree which is a BST.Info largestBSTBT(Node* root){    // Base cases : When tree is empty or it has    // one child.    if (root == NULL)        return {0, INT_MIN, INT_MAX, 0, true};    if (root->left == NULL && root->right == NULL)        return {1, root->data, root->data, 1, true};     // Recur for left subtree and right subtrees    Info l = largestBSTBT(root->left);    Info r = largestBSTBT(root->right);     // Create a return variable and initialize its    // size.    Info ret;    ret.sz = (1 + l.sz + r.sz);     // If whole tree rooted under current root is    // BST.    if (l.isBST && r.isBST && l.max < root->data &&            r.min > root->data)    {        ret.min = min(l.min, min(r.min, root->data));        ret.max = max(r.max, max(l.max, root->data));         // Update answer for tree rooted under        // current 'root'        ret.ans = ret.sz;        ret.isBST = true;         return ret;    }     // If whole tree is not BST, return maximum    // of left and right subtrees    ret.ans = max(l.ans, r.ans);    ret.isBST = false;     return ret;} /* Driver program to test above functions*/int main(){    /* Let us construct the following Tree        60       /  \      65  70     /    50 */     struct Node *root = newNode(60);    root->left = newNode(65);    root->right = newNode(70);    root->left->left = newNode(50);    printf(" Size of the largest BST is %d\n",           largestBSTBT(root).ans);    return 0;} // This code is contributed by Vivek Garg in a// comment on below set 1.// www.geeksforgeeks.org/find-the-largest-subtree-in-a-tree-that-is-also-a-bst/

Java

 // Java program to find largest BST in a Binary Tree. /* A binary tree node has data, pointer to left child and a pointer to right child */class Node {  int data;  Node left, right;   public Node(final int d) {    data = d;  }} class GFG {   public static void main(String[] args) {        /* Let us construct the following Tree        60       /  \      65  70     /    50 */     final Node node1 = new Node(60);    node1.left = new Node(65);    node1.right = new Node(70);    node1.left.left = new Node(50);        System.out.print("Size of the largest BST is " + Solution.largestBst(node1) + "\n");    }} class Solution {  static int MAX = Integer.MAX_VALUE;  static int MIN = Integer.MIN_VALUE;   static class nodeInfo {    int size; // Size of subtree    int max; // Min value in subtree    int min; // Max value in subtree    boolean isBST; // If subtree is BST     nodeInfo() {    }     nodeInfo(int size, int max, int min, boolean isBST) {      this.size = size;      this.max = max;      this.min = min;      this.isBST = isBST;    }  }   static nodeInfo largestBST(Node root) {     // Base cases : When tree is empty or it has one child.    if (root == null) {      return new nodeInfo(0, MIN, MAX, true);    }    if (root.left == null && root.right == null) {      return new nodeInfo(1, root.data, root.data, true);    }     // Recur for left subtree and right subtrees    nodeInfo left = largestBST(root.left);    nodeInfo right = largestBST(root.right);     // Create a return variable and initialize its size.    nodeInfo returnInfo = new nodeInfo();     // If whole tree rooted under current root is BST.    if (left.isBST && right.isBST && left.max < root.data && right.min > root.data) {      returnInfo.min = Math.min(Math.min(left.min, right.min), root.data);      returnInfo.max = Math.max(Math.max(left.max, right.max), root.data);       // Update answer for tree rooted under current 'root'      returnInfo.size = left.size + 1 + right.size;      returnInfo.isBST = true;      return returnInfo;    }     // If whole tree is not BST, return maximum of left and right subtrees    returnInfo.size = Math.max(left.size, right.size);    returnInfo.isBST = false;    return returnInfo;  }   // Return the size of the largest sub-tree which is also a BST  static int largestBst(Node root) {    return largestBST(root).size;  }}// This code is contributed by Andrei Sljusar

Python3

 # Python program to find largest# BST in a Binary Tree. INT_MIN = -2147483648INT_MAX = 2147483647 # Helper function that allocates a new# node with the given data and None left# and right pointers.class newNode:     # Constructor to create a new node    def __init__(self, data):        self.data = data        self.left = None        self.right = None # Returns Information about subtree. The# Information also includes size of largest# subtree which is a BSTdef largestBSTBT(root):     # Base cases : When tree is empty or it has    # one child.    if (root == None):        return 0, INT_MIN, INT_MAX, 0, True    if (root.left == None and root.right == None) :        return 1, root.data, root.data, 1, True     # Recur for left subtree and right subtrees    l = largestBSTBT(root.left)    r = largestBSTBT(root.right)     # Create a return variable and initialize its    # size.    ret = [0, 0, 0, 0, 0]    ret = (1 + l + r)     # If whole tree rooted under current root is    # BST.    if (l and r and l <        root.data and r > root.data) :             ret = min(l, min(r, root.data))        ret = max(r, max(l, root.data))         # Update answer for tree rooted under        # current 'root'        ret = ret        ret = True         return ret          # If whole tree is not BST, return maximum    # of left and right subtrees    ret = max(l, r)    ret = False     return ret # Driver Codeif __name__ == '__main__':         """Let us construct the following Tree        60        / \        65 70    /    50 """    root = newNode(60)    root.left = newNode(65)    root.right = newNode(70)    root.left.left = newNode(50)    print("Size of the largest BST is",                    largestBSTBT(root))                             # This code is contributed# Shubham Singh(SHUBHAMSINGH10)
Output
Size of the largest BST is 2

Time Complexity : O(n)
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