Lowest Common Ancestor in a Binary Tree using Parent Pointer
Given values of two nodes in a Binary Tree, find the Lowest Common Ancestor (LCA). It may be assumed that both nodes exist in the tree.
For example, consider the Binary Tree in diagram, LCA of 10 and 14 is 12 and LCA of 8 and 14 is 8.
Let T be a rooted tree. The lowest common ancestor between two nodes n1 and n2 is defined as the lowest node in T that has both n1 and n2 as descendants (where we allow a node to be a descendant of itself). Source : Wikipedia.
We have discussed different approaches to find LCA in set 1. Finding LCA becomes easy when parent pointer is given as we can easily find all ancestors of a node using parent pointer. Below are steps to find LCA.
- Create an empty hash table.
- Insert n1 and all of its ancestors in hash table.
- Check if n2 or any of its ancestors exist in hash table, if yes return the first existing ancestor.
Below is the implementation of above steps.
C++14
// C++ program to find lowest common ancestor using parent pointer#include <bits/stdc++.h>using namespace std;// A Tree Nodestruct Node{ Node *left, *right, *parent; int key;};// A utility function to create a new BST nodeNode *newNode(int item){ Node *temp = new Node; temp->key = item; temp->parent = temp->left = temp->right = NULL; return temp;}/* A utility function to insert a new node with given key in Binary Search Tree */Node *insert(Node *node, int key){ /* If the tree is empty, return a new node */ if (node == NULL) return newNode(key); /* Otherwise, recur down the tree */ if (key < node->key) { node->left = insert(node->left, key); node->left->parent = node; } else if (key > node->key) { node->right = insert(node->right, key); node->right->parent = node; } /* return the (unchanged) node pointer */ return node;}// To find LCA of nodes n1 and n2 in Binary TreeNode *LCA(Node *n1, Node *n2){ // Create a map to store ancestors of n1 map <Node *, bool> ancestors; // Insert n1 and all its ancestors in map while (n1 != NULL) { ancestors[n1] = true; n1 = n1->parent; } // Check if n2 or any of its ancestors is in // map. while (n2 != NULL) { if (ancestors.find(n2) != ancestors.end()) return n2; n2 = n2->parent; } return NULL;}// Driver method to test above functionsint main(void){ Node * root = NULL; root = insert(root, 20); root = insert(root, 8); root = insert(root, 22); root = insert(root, 4); root = insert(root, 12); root = insert(root, 10); root = insert(root, 14); Node *n1 = root->left->right->left; Node *n2 = root->left; Node *lca = LCA(n1, n2); printf("LCA of %d and %d is %d \n", n1->key, n2->key, lca->key); return 0;} |
Java
import java.util.HashMap;import java.util.Map;// Java program to find lowest common ancestor using parent pointer// A tree nodeclass Node{ int key; Node left, right, parent; Node(int key) { this.key = key; left = right = parent = null; }}class BinaryTree{ Node root, n1, n2, lca; /* A utility function to insert a new node with given key in Binary Search Tree */ Node insert(Node node, int key) { /* If the tree is empty, return a new node */ if (node == null) return new Node(key); /* Otherwise, recur down the tree */ if (key < node.key) { node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key) { node.right = insert(node.right, key); node.right.parent = node; } /* return the (unchanged) node pointer */ return node; } // To find LCA of nodes n1 and n2 in Binary Tree Node LCA(Node n1, Node n2) { // Create a map to store ancestors of n1 Map<Node, Boolean> ancestors = new HashMap<Node, Boolean>(); // Insert n1 and all its ancestors in map while (n1 != null) { ancestors.put(n1, Boolean.TRUE); n1 = n1.parent; } // Check if n2 or any of its ancestors is in // map. while (n2 != null) { if (ancestors.containsKey(n2) != ancestors.isEmpty()) return n2; n2 = n2.parent; } return null; } // Driver method to test above functions public static void main(String[] args) { BinaryTree tree = new BinaryTree(); tree.root = tree.insert(tree.root, 20); tree.root = tree.insert(tree.root, 8); tree.root = tree.insert(tree.root, 22); tree.root = tree.insert(tree.root, 4); tree.root = tree.insert(tree.root, 12); tree.root = tree.insert(tree.root, 10); tree.root = tree.insert(tree.root, 14); tree.n1 = tree.root.left.right.left; tree.n2 = tree.root.left; tree.lca = tree.LCA(tree.n1, tree.n2); System.out.println("LCA of " + tree.n1.key + " and " + tree.n2.key + " is " + tree.lca.key); }}// This code has been contributed by Mayank Jaiswal(mayank_24) |
Python3
# Python program to find lowest common ancestor using parent pointerglobal rootroot = None# A Tree Nodeclass Node: def __init__(self, key): self.key = key self.left = None self.right = None self.parent = None# A utility function to insert a new node with given key in Binary Search Treedef insert(root, key): # If the tree is empty, return a new node if root == None: return Node(key) # Otherwise, recur down the tree if key < root.key: root.left = insert(root.left, key) root.left.parent = root elif key > root.key: root.right = insert(root.right, key) root.right.parent = root # return the (unchanged) node pointer return root# To find LCA of nodes n1 and n2 in Binary Treedef LCA(n1, n2): # Create a map to store ancestors of n1 ancestors = {} # Insert n1 and all its ancestors in map while n1 != None: ancestors[n1] = True n1 = n1.parent # Check if n2 or any of its ancestors is in map. while n2 != None: if n2 in ancestors: return n2 n2 = n2.parent return None# Driver method to test above functionsif __name__ == '__main__': root = insert(root, 20) root = insert(root, 8) root = insert(root, 22) root = insert(root, 4) root = insert(root, 12) root = insert(root, 10) root = insert(root, 14) n1 = root.left.right.left n2 = root.left lca = LCA(n1, n2) print("LCA of", n1.key, "and", n2.key, "is", lca.key)# This code is contributed by Tapesh(tapeshdua420) |
C#
// C# program to find lowest common ancestor using parent pointer// A tree nodeusing System;using System.Collections;using System.Collections.Generic;public class Node{ public int key; public Node left, right, parent; public Node(int key) { this.key = key; left = right = parent = null; }}class BinaryTree{ Node root, n1, n2, lca; /* A utility function to insert a new node with given key in Binary Search Tree */ Node insert(Node node, int key) { /* If the tree is empty, return a new node */ if (node == null) return new Node(key); /* Otherwise, recur down the tree */ if (key < node.key) { node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key) { node.right = insert(node.right, key); node.right.parent = node; } /* return the (unchanged) node pointer */ return node; } // To find LCA of nodes n1 and n2 in Binary Tree Node LCA(Node n1, Node n2) { // Create a map to store ancestors of n1 Dictionary<Node, Boolean> ancestors = new Dictionary<Node, Boolean>(); // Insert n1 and all its ancestors in map while (n1 != null) { ancestors.Add(n1,true); n1 = n1.parent; } // Check if n2 or any of its ancestors is in // map. while (n2 != null) { if (ancestors.ContainsKey(n2)) return n2; n2 = n2.parent; } return null; } // Driver code public static void Main(String []args) { BinaryTree tree = new BinaryTree(); tree.root = tree.insert(tree.root, 20); tree.root = tree.insert(tree.root, 8); tree.root = tree.insert(tree.root, 22); tree.root = tree.insert(tree.root, 4); tree.root = tree.insert(tree.root, 12); tree.root = tree.insert(tree.root, 10); tree.root = tree.insert(tree.root, 14); tree.n1 = tree.root.left.right.left; tree.n2 = tree.root.left; tree.lca = tree.LCA(tree.n1, tree.n2); Console.WriteLine("LCA of " + tree.n1.key + " and " + tree.n2.key + " is " + tree.lca.key); }}// This code is contributed by Arnab Kundu |
Javascript
// Javascript code for above approach// A Tree Nodeclass Node { constructor(left, right, parent, key) { this.left = left; this.right = right; this.parent = parent; this.key = key; }}// A utility function to create a new BST nodefunction newNode(item) { let temp = new Node(); temp.key = item; temp.parent = temp.left = temp.right = null; return temp;}/* A utility function to insert a new node withgiven key in Binary Search Tree */function insert(node, key) { /* If the tree is empty, return a new node */ if (node == null) return newNode(key); /* Otherwise, recur down the tree */ if (key < node.key) { node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key) { node.right = insert(node.right, key); node.right.parent = node; } /* return the (unchanged) node pointer */ return node;}// To find LCA of nodes n1 and n2 in Binary Treefunction LCA(n1, n2) { // Create a map to store ancestors of n1 let ancestors = new Map(); // Insert n1 and all its ancestors in map while (n1 != null) { ancestors.set(n1, true); n1 = n1.parent; } // Check if n2 or any of its ancestors is in // map. while (n2 != null) { if (ancestors.has(n2)) return n2; n2 = n2.parent; } return null;}// Driver method to test above functionslet root = null;root = insert(root, 20);root = insert(root, 8);root = insert(root, 22);root = insert(root, 4);root = insert(root, 12);root = insert(root, 10);root = insert(root, 14);let n1 = root.left.right.left;let n2 = root.left;let lca = LCA(n1, n2);console.log(`LCA of ${n1.key} and ${n2.key} is ${lca.key}`);// This code is contributed by adityamaharshi21 |
Output:
LCA of 10 and 8 is 8
Note : The above implementation uses insert of Binary Search Tree to create a Binary Tree, but the function LCA is for any Binary Tree (not necessarily a Binary Search Tree). Time Complexity : O(h) where h is height of Binary Tree if we use hash table to implement the solution (Note that the above solution uses map which takes O(Log h) time to insert and find). So the time complexity of above implementation is O(h Log h). Auxiliary Space : O(h) A O(h) time and O(1) Extra Space Solution: The above solution requires extra space because we need to use a hash table to store visited ancestors. We can solve the problem in O(1) extra space using following fact : If both nodes are at same level and if we traverse up using parent pointers of both nodes, the first common node in the path to root is lca. The idea is to find depths of given nodes and move up the deeper node pointer by the difference between depths. Once both nodes reach same level, traverse them up and return the first common node. Thanks to Mysterious Mind for suggesting this approach.
C++
// C++ program to find lowest common ancestor using parent pointer#include <bits/stdc++.h>using namespace std;// A Tree Nodestruct Node{ Node *left, *right, *parent; int key;};// A utility function to create a new BST nodeNode *newNode(int item){ Node *temp = new Node; temp->key = item; temp->parent = temp->left = temp->right = NULL; return temp;}/* A utility function to insert a new node withgiven key in Binary Search Tree */Node *insert(Node *node, int key){ /* If the tree is empty, return a new node */ if (node == NULL) return newNode(key); /* Otherwise, recur down the tree */ if (key < node->key) { node->left = insert(node->left, key); node->left->parent = node; } else if (key > node->key) { node->right = insert(node->right, key); node->right->parent = node; } /* return the (unchanged) node pointer */ return node;}// A utility function to find depth of a node// (distance of it from root)int depth(Node *node){ int d = -1; while (node) { ++d; node = node->parent; } return d;}// To find LCA of nodes n1 and n2 in Binary TreeNode *LCA(Node *n1, Node *n2){ // Find depths of two nodes and differences int d1 = depth(n1), d2 = depth(n2); int diff = d1 - d2; // If n2 is deeper, swap n1 and n2 if (diff < 0) { Node * temp = n1; n1 = n2; n2 = temp; diff = -diff; } // Move n1 up until it reaches the same level as n2 while (diff--) n1 = n1->parent; // Now n1 and n2 are at same levels while (n1 && n2) { if (n1 == n2) return n1; n1 = n1->parent; n2 = n2->parent; } return NULL;}// Driver method to test above functionsint main(void){ Node * root = NULL; root = insert(root, 20); root = insert(root, 8); root = insert(root, 22); root = insert(root, 4); root = insert(root, 12); root = insert(root, 10); root = insert(root, 14); Node *n1 = root->left->right->left; Node *n2 = root->right; Node *lca = LCA(n1, n2); printf("LCA of %d and %d is %d \n", n1->key, n2->key, lca->key); return 0;} |
Java
import java.util.HashMap;import java.util.Map;// Java program to find lowest common ancestor using parent pointer// A tree nodeclass Node{ int key; Node left, right, parent; Node(int key) { this.key = key; left = right = parent = null; }}class BinaryTree{ Node root, n1, n2, lca; /* A utility function to insert a new node with given key in Binary Search Tree */ Node insert(Node node, int key) { /* If the tree is empty, return a new node */ if (node == null) return new Node(key); /* Otherwise, recur down the tree */ if (key < node.key) { node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key) { node.right = insert(node.right, key); node.right.parent = node; } /* return the (unchanged) node pointer */ return node; } // A utility function to find depth of a node // (distance of it from root) int depth(Node node) { int d = -1; while (node != null) { ++d; node = node.parent; } return d; } // To find LCA of nodes n1 and n2 in Binary Tree Node LCA(Node n1, Node n2) { // Find depths of two nodes and differences int d1 = depth(n1), d2 = depth(n2); int diff = d1 - d2; // If n2 is deeper, swap n1 and n2 if (diff < 0) { Node temp = n1; n1 = n2; n2 = temp; diff = -diff; } // Move n1 up until it reaches the same level as n2 while (diff-- != 0) n1 = n1.parent; // Now n1 and n2 are at same levels while (n1 != null && n2 != null) { if (n1 == n2) return n1; n1 = n1.parent; n2 = n2.parent; } return null; } // Driver method to test above functions public static void main(String[] args) { BinaryTree tree = new BinaryTree(); tree.root = tree.insert(tree.root, 20); tree.root = tree.insert(tree.root, 8); tree.root = tree.insert(tree.root, 22); tree.root = tree.insert(tree.root, 4); tree.root = tree.insert(tree.root, 12); tree.root = tree.insert(tree.root, 10); tree.root = tree.insert(tree.root, 14); tree.n1 = tree.root.left.right.left; tree.n2 = tree.root.right; tree.lca = tree.LCA(tree.n1, tree.n2); System.out.println("LCA of " + tree.n1.key + " and " + tree.n2.key + " is " + tree.lca.key); }}// This code has been contributed by Mayank Jaiswal(mayank_24) |
Python3
# Python program to find lowest common ancestor using parent pointer# A tree nodeclass Node: def __init__(self, key): self.key = key self.left = None self.right = None self.parent = Noneclass BinaryTree: def __init__(self): self.root = None self.n1 = None self.n2 = None self.lca = None # A utility function to insert a new node with # given key in Binary Search Tree def insert(self, node, key): # If the tree is empty, return a new node if node == None: return Node(key) # Otherwise, recur down the tree if key < node.key: node.left = self.insert(node.left, key) node.left.parent = node elif key > node.key: node.right = self.insert(node.right, key) node.right.parent = node # return the (unchanged) node pointer return node # A utility function to find depth of a node (distance of it from root) def depth(self, node): d = -1 while(node != None): d += 1 node = node.parent return d # To find LCA of nodes n1 and n2 in Binary Tree def LCA(self, n1, n2): # Find depths of two nodes and differences d1 = self.depth(n1) d2 = self.depth(n2) diff = d1-d2 # If n2 is deeper, swap n1 and n2 if diff < 0: temp = n1 n1 = n2 n2 = temp diff = -diff # Move n1 up until it reaches the same level as n2 while diff != 0: n1 = n1.parent diff -= 1 # Now n1 and n2 are at same levels while n1 != None and n2 != None: if (n1 == n2): return n1 n1 = n1.parent n2 = n2.parent return None# Driver method to test above functionsif __name__ == '__main__': tree = BinaryTree() tree.root = tree.insert(tree.root, 20) tree.root = tree.insert(tree.root, 8) tree.root = tree.insert(tree.root, 22) tree.root = tree.insert(tree.root, 4) tree.root = tree.insert(tree.root, 12) tree.root = tree.insert(tree.root, 10) tree.root = tree.insert(tree .root, 14) tree.n1 = tree.root.left.right.left tree.n2 = tree.root.right tree.lca = tree.LCA(tree.n1, tree.n2) print("LCA of " + str(tree.n1.key) + " and " + str(tree.n2.key) + " is " + str(tree .lca .key))# This code is contributed by Tapesh(tapesh1308) |
C#
// C# program to find lowest common// ancestor using parent pointerusing System;// A tree nodepublic class Node{ public int key; public Node left, right, parent; public Node(int key) { this.key = key; left = right = parent = null; }}class GFG{public Node root, n1, n2, lca;/* A utility function to insert a new node with given key in Binary Search Tree */public virtual Node insert(Node node, int key){ /* If the tree is empty, return a new node */ if (node == null) { return new Node(key); } /* Otherwise, recur down the tree */ if (key < node.key) { node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key) { node.right = insert(node.right, key); node.right.parent = node; } /* return the (unchanged) node pointer */ return node;}// A utility function to find depth of a// node (distance of it from root)public virtual int depth(Node node){ int d = -1; while (node != null) { ++d; node = node.parent; } return d;}// To find LCA of nodes n1 and n2// in Binary Treepublic virtual Node LCA(Node n1, Node n2){ // Find depths of two nodes // and differences int d1 = depth(n1), d2 = depth(n2); int diff = d1 - d2; // If n2 is deeper, swap n1 and n2 if (diff < 0) { Node temp = n1; n1 = n2; n2 = temp; diff = -diff; } // Move n1 up until it reaches // the same level as n2 while (diff-- != 0) { n1 = n1.parent; } // Now n1 and n2 are at same levels while (n1 != null && n2 != null) { if (n1 == n2) { return n1; } n1 = n1.parent; n2 = n2.parent; } return null;}// Driver Codepublic static void Main(string[] args){ GFG tree = new GFG(); tree.root = tree.insert(tree.root, 20); tree.root = tree.insert(tree.root, 8); tree.root = tree.insert(tree.root, 22); tree.root = tree.insert(tree.root, 4); tree.root = tree.insert(tree.root, 12); tree.root = tree.insert(tree.root, 10); tree.root = tree.insert(tree.root, 14); tree.n1 = tree.root.left.right.left; tree.n2 = tree.root.right; tree.lca = tree.LCA(tree.n1, tree.n2); Console.WriteLine("LCA of " + tree.n1.key + " and " + tree.n2.key + " is " + tree.lca.key);}}// This code is contributed by Shrikant13 |
Javascript
// Javascript code for above approach // A Tree Nodeclass Node { constructor(left, right, parent, key) { this.left = left; this.right = right; this.parent = parent; this.key = key; }}// A utility function to create a new BST nodefunction newNode(item) { let temp = new Node(); temp.key = item; temp.parent = temp.left = temp.right = null; return temp;}// A utility function to insert a new node with// given key in Binary Search Treefunction insert(node, key){ // If the tree is empty, return a new node if (node == null) return newNode(key); // Otherwise, recur down the tree if (key < node.key){ node.left = insert(node.left, key); node.left.parent = node; } else if (key > node.key){ node.right = insert(node.right, key); node.right.parent = node; } // return the (unchanged) node pointer return node;}// A utility function to find depth of a node// (distance of it from root)function depth(node){ let d = -1; while(node){ ++d; node = node.parent; } return d;}// To find LCA of nodes n1 and n2 in Binary Treefunction LCA(n1, n2){ // Find depths of two nodes and differences let d1 = depth(n1), d2 = depth(n2); let diff = d1 - d2; // If n2 is deeper, swap n1 and n2 if (diff < 0){ let temp = n1; n1 = n2; n2 = temp; diff = -diff; } // Move n1 up until it reaches the same level as n2 while (diff--) n1 = n1.parent; // Now n1 and n2 are at same levels while (n1 && n2){ if (n1 == n2) return n1; n1 = n1.parent; n2 = n2.parent; } return null;}// Driver method to test above functionslet root = null; root = insert(root, 20);root = insert(root, 8);root = insert(root, 22);root = insert(root, 4);root = insert(root, 12);root = insert(root, 10);root = insert(root, 14); let n1 = root.left.right.left;let n2 = root.right;let lca = LCA(n1, n2);console.log(`LCA of ${n1.key} and ${n2.key} is ${lca.key}`);// this code is contributed by Yash Agarwal(yashagarwal2852002) |
Output :
LCA of 10 and 22 is 20
Time Complexity : O(h)
Space Complexity : O(1)
You may like to see below articles as well : Lowest Common Ancestor in a Binary Tree | Set 1 Lowest Common Ancestor in a Binary Search Tree. Find LCA in Binary Tree using RMQ This article is contributed by Dheeraj Gupta. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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