Given a binary tree (not a binary search tree) and two values say n1 and n2, write a program to find the least common ancestor.

**Following is definition of LCA from Wikipedia:**

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).

The LCA of n1 and n2 in T is the shared ancestor of n1 and n2 that is located farthest from the root. Computation of lowest common ancestors may be useful, for instance, as part of a procedure for determining the distance between pairs of nodes in a tree: the distance from n1 to n2 can be computed as the distance from the root to n1, plus the distance from the root to n2, minus twice the distance from the root to their lowest common ancestor. (Source Wiki)

We have discussed an efficient solution to find LCA in Binary Search Tree. In Binary Search Tree, using BST properties, we can find LCA in O(h) time where h is height of tree. Such an implementation is not possible in Binary Tree as keys Binary Tree nodes don’t follow any order. Following are different approaches to find LCA in Binary Tree.

**Method 1 (By Storing root to n1 and root to n2 paths):**

Following is simple O(n) algorithm to find LCA of n1 and n2.

**1) **Find path from root to n1 and store it in a vector or array.

**2) **Find path from root to n2 and store it in another vector or array.

**3)** Traverse both paths till the values in arrays are same. Return the common element just before the mismatch.

Following is C++ implementation of above algorithm.

## C++

// A O(n) solution to find LCA of two given values n1 and n2 #include <iostream> #include <vector> using namespace std; // A Bianry Tree node struct Node { int key; struct Node *left, *right; }; // Utility function creates a new binary tree node with given key Node * newNode(int k) { Node *temp = new Node; temp->key = k; temp->left = temp->right = NULL; return temp; } // Finds the path from root node to given root of the tree, Stores the // path in a vector path[], returns true if path exists otherwise false bool findPath(Node *root, vector<int> &path, int k) { // base case if (root == NULL) return false; // Store this node in path vector. The node will be removed if // not in path from root to k path.push_back(root->key); // See if the k is same as root's key if (root->key == k) return true; // Check if k is found in left or right sub-tree if ( (root->left && findPath(root->left, path, k)) || (root->right && findPath(root->right, path, k)) ) return true; // If not present in subtree rooted with root, remove root from // path[] and return false path.pop_back(); return false; } // Returns LCA if node n1, n2 are present in the given binary tree, // otherwise return -1 int findLCA(Node *root, int n1, int n2) { // to store paths to n1 and n2 from the root vector<int> path1, path2; // Find paths from root to n1 and root to n1. If either n1 or n2 // is not present, return -1 if ( !findPath(root, path1, n1) || !findPath(root, path2, n2)) return -1; /* Compare the paths to get the first different value */ int i; for (i = 0; i < path1.size() && i < path2.size() ; i++) if (path1[i] != path2[i]) break; return path1[i-1]; } // Driver program to test above functions int main() { // Let us create the Binary Tree shown in above diagram. Node * root = newNode(1); root->left = newNode(2); root->right = newNode(3); root->left->left = newNode(4); root->left->right = newNode(5); root->right->left = newNode(6); root->right->right = newNode(7); cout << "LCA(4, 5) = " << findLCA(root, 4, 5); cout << "\nLCA(4, 6) = " << findLCA(root, 4, 6); cout << "\nLCA(3, 4) = " << findLCA(root, 3, 4); cout << "\nLCA(2, 4) = " << findLCA(root, 2, 4); return 0; }

## Python

# O(n) solution to find LCS of two given values n1 and n2 # A binary tree node class Node: # Constructor to create a new binary node def __init__(self, key): self.key = key self.left = None self.right = None # Finds the path from root node to given root of the tree. # Stores the path in a list path[], returns true if path # exists otherwise false def findPath( root, path, k): # Baes Case if root is None: return False # Store this node is path vector. The node will be # removed if not in path from root to k path.append(root.key) # See if the k is same as root's key if root.key == k : return True # Check if k is found in left or right sub-tree if ((root.left != None and findPath(root.left, path, k)) or (root.right!= None and findPath(root.right, path, k))): return True # If not present in subtree rooted with root, remove # root from path and return False path.pop() return False # Returns LCA if node n1 , n2 are present in the given # binary tre otherwise return -1 def findLCA(root, n1, n2): # To store paths to n1 and n2 fromthe root path1 = [] path2 = [] # Find paths from root to n1 and root to n2. # If either n1 or n2 is not present , return -1 if (not findPath(root, path1, n1) or not findPath(root, path2, n2)): return -1 # Compare the paths to get the first different value i = 0 while(i < len(path1) and i < len(path2)): if path1[i] != path2[i]: break i += 1 return path1[i-1] # Driver program to test above function # Let's create the Binary Tree shown in above diagram root = Node(1) root.left = Node(2) root.right = Node(3) root.left.left = Node(4) root.left.right = Node(5) root.right.left = Node(6) root.right.right = Node(7) print "LCA(4, 5) = %d" %(findLCA(root, 4, 5,)) print "LCA(4, 6) = %d" %(findLCA(root, 4, 6)) print "LCA(3, 4) = %d" %(findLCA(root,3,4)) print "LCA(2, 4) = %d" %(findLCA(root,2, 4)) # This code is contributed by Nikhil Kumar Singh(nickzuck_007)

Output:

LCA(4, 5) = 2 LCA(4, 6) = 1 LCA(3, 4) = 1 LCA(2, 4) = 2

* Time Complexity: *Time complexity of the above solution is O(n). The tree is traversed twice, and then path arrays are compared.

Thanks to

*Ravi Chandra Enaganti*for suggesting the initial solution based on this method.

**Method 2 (Using Single Traversal)**

The method 1 finds LCA in O(n) time, but requires three tree traversals plus extra spaces for path arrays. If we assume that the keys n1 and n2 are present in Binary Tree, we can find LCA using single traversal of Binary Tree and without extra storage for path arrays.

The idea is to traverse the tree starting from root. If any of the given keys (n1 and n2) matches with root, then root is LCA (assuming that both keys are present). If root doesn’t match with any of the keys, we recur for left and right subtree. The node which has one key present in its left subtree and the other key present in right subtree is the LCA. If both keys lie in left subtree, then left subtree has LCA also, otherwise LCA lies in right subtree.

## C

/* Program to find LCA of n1 and n2 using one traversal of Binary Tree */ #include <iostream> using namespace std; // A Binary Tree Node struct Node { struct Node *left, *right; int key; }; // Utility function to create a new tree Node Node* newNode(int key) { Node *temp = new Node; temp->key = key; temp->left = temp->right = NULL; return temp; } // This function returns pointer to LCA of two given values n1 and n2. // This function assumes that n1 and n2 are present in Binary Tree struct Node *findLCA(struct Node* root, int n1, int n2) { // Base case if (root == NULL) return NULL; // If either n1 or n2 matches with root's key, report // the presence by returning root (Note that if a key is // ancestor of other, then the ancestor key becomes LCA if (root->key == n1 || root->key == n2) return root; // Look for keys in left and right subtrees Node *left_lca = findLCA(root->left, n1, n2); Node *right_lca = findLCA(root->right, n1, n2); // If both of the above calls return Non-NULL, then one key // is present in once subtree and other is present in other, // So this node is the LCA if (left_lca && right_lca) return root; // Otherwise check if left subtree or right subtree is LCA return (left_lca != NULL)? left_lca: right_lca; } // Driver program to test above functions int main() { // Let us create binary tree given in the above example Node * root = newNode(1); root->left = newNode(2); root->right = newNode(3); root->left->left = newNode(4); root->left->right = newNode(5); root->right->left = newNode(6); root->right->right = newNode(7); cout << "LCA(4, 5) = " << findLCA(root, 4, 5)->key; cout << "\nLCA(4, 6) = " << findLCA(root, 4, 6)->key; cout << "\nLCA(3, 4) = " << findLCA(root, 3, 4)->key; cout << "\nLCA(2, 4) = " << findLCA(root, 2, 4)->key; return 0; }

## Java

//Java implementation to find lowest common ancestor of // n1 and n2 using one traversal of binary tree /* Class containing left and right child of current node and key value*/ class Node { int data; Node left, right; public Node(int item) { data = item; left = right = null; } } public class BinaryTree { //Root of the Binary Tree Node root; Node findLCA(int n1, int n2) { return findLCA(root, n1, n2); } // This function returns pointer to LCA of two given // values n1 and n2. This function assumes that n1 and // n2 are present in Binary Tree Node findLCA(Node node, int n1, int n2) { // Base case if (node == null) return null; // If either n1 or n2 matches with root's key, report // the presence by returning root (Note that if a key is // ancestor of other, then the ancestor key becomes LCA if (node.data == n1 || node.data == n2) return node; // Look for keys in left and right subtrees Node left_lca = findLCA(node.left, n1, n2); Node right_lca = findLCA(node.right, n1, n2); // If both of the above calls return Non-NULL, then one key // is present in once subtree and other is present in other, // So this node is the LCA if (left_lca!=null && right_lca!=null) return node; // Otherwise check if left subtree or right subtree is LCA return (left_lca != null) ? left_lca : right_lca; } /* Driver program to test above functions */ public static void main(String args[]) { BinaryTree tree = new BinaryTree(); tree.root = new Node(1); tree.root.left = new Node(2); tree.root.right = new Node(3); tree.root.left.left = new Node(4); tree.root.left.right = new Node(5); tree.root.right.left = new Node(6); tree.root.right.right = new Node(7); System.out.println("LCA(4, 5) = " + tree.findLCA(4, 5).data); System.out.println("LCA(4, 6) = " + tree.findLCA(4, 6).data); System.out.println("LCA(3, 4) = " + tree.findLCA(3, 4).data); System.out.println("LCA(2, 4) = " + tree.findLCA(2, 4).data); } }

## Python

# Python program to find LCA of n1 and n2 using one # traversal of Binary tree # A binary tree node class Node: # Constructor to create a new tree node def __init__(self, key): self.key = key self.left = None self.right = None # This function returns pointer to LCA of two given # values n1 and n2 # This function assumes that n1 and n2 are present in # Binary Tree def findLCA(root, n1, n2): # Base Case if root is None: return None # If either n1 or n2 matches with root's key, report # the presence by returning root (Note that if a key is # ancestor of other, then the ancestor key becomes LCA if root.key == n1 or root.key == n2: return root # Look for keys in left and right subtrees left_lca = findLCA(root.left, n1, n2) right_lca = findLCA(root.right, n1, n2) # If both of the above calls return Non-NULL, then one key # is present in once subtree and other is present in other, # So this node is the LCA if left_lca and right_lca: return root # Otherwise check if left subtree or right subtree is LCA return left_lca if left_lca is not None else right_lca # Driver program to test above function # Let us create a binary tree given in the above example root = Node(1) root.left = Node(2) root.right = Node(3) root.left.left = Node(4) root.left.right = Node(5) root.right.left = Node(6) root.right.right = Node(7) print "LCA(4,5) = ", findLCA(root, 4, 5).key print "LCA(4,6) = ", findLCA(root, 4, 6).key print "LCA(3,4) = ", findLCA(root, 3, 4).key print "LCA(2,4) = ", findLCA(root, 2, 4).key # This code is contributed by Nikhil Kumar Singh(nickzuck_007)

Output:

LCA(4, 5) = 2 LCA(4, 6) = 1 LCA(3, 4) = 1 LCA(2, 4) = 2

Thanks to *Atul Singh* for suggesting this solution.

* Time Complexity: *Time complexity of the above solution is O(n) as the method does a simple tree traversal in bottom up fashion.

Note that the above method assumes that keys are present in Binary Tree. If one key is present and other is absent, then it returns the present key as LCA (Ideally should have returned NULL).

We can extend this method to handle all cases by passing two boolean variables v1 and v2. v1 is set as true when n1 is present in tree and v2 is set as true if n2 is present in tree.

## c

/* Program to find LCA of n1 and n2 using one traversal of Binary Tree. It handles all cases even when n1 or n2 is not there in Binary Tree */ #include <iostream> using namespace std; // A Binary Tree Node struct Node { struct Node *left, *right; int key; }; // Utility function to create a new tree Node Node* newNode(int key) { Node *temp = new Node; temp->key = key; temp->left = temp->right = NULL; return temp; } // This function returns pointer to LCA of two given values n1 and n2. // v1 is set as true by this function if n1 is found // v2 is set as true by this function if n2 is found struct Node *findLCAUtil(struct Node* root, int n1, int n2, bool &v1, bool &v2) { // Base case if (root == NULL) return NULL; // If either n1 or n2 matches with root's key, report the presence // by setting v1 or v2 as true and return root (Note that if a key // is ancestor of other, then the ancestor key becomes LCA) if (root->key == n1) { v1 = true; return root; } if (root->key == n2) { v2 = true; return root; } // Look for keys in left and right subtrees Node *left_lca = findLCAUtil(root->left, n1, n2, v1, v2); Node *right_lca = findLCAUtil(root->right, n1, n2, v1, v2); // If both of the above calls return Non-NULL, then one key // is present in once subtree and other is present in other, // So this node is the LCA if (left_lca && right_lca) return root; // Otherwise check if left subtree or right subtree is LCA return (left_lca != NULL)? left_lca: right_lca; } // Returns true if key k is present in tree rooted with root bool find(Node *root, int k) { // Base Case if (root == NULL) return false; // If key is present at root, or in left subtree or right subtree, // return true; if (root->key == k || find(root->left, k) || find(root->right, k)) return true; // Else return false return false; } // This function returns LCA of n1 and n2 only if both n1 and n2 are present // in tree, otherwise returns NULL; Node *findLCA(Node *root, int n1, int n2) { // Initialize n1 and n2 as not visited bool v1 = false, v2 = false; // Find lca of n1 and n2 using the technique discussed above Node *lca = findLCAUtil(root, n1, n2, v1, v2); // Return LCA only if both n1 and n2 are present in tree if (v1 && v2 || v1 && find(lca, n2) || v2 && find(lca, n1)) return lca; // Else return NULL return NULL; } // Driver program to test above functions int main() { // Let us create binary tree given in the above example Node * root = newNode(1); root->left = newNode(2); root->right = newNode(3); root->left->left = newNode(4); root->left->right = newNode(5); root->right->left = newNode(6); root->right->right = newNode(7); Node *lca = findLCA(root, 4, 5); if (lca != NULL) cout << "LCA(4, 5) = " << lca->key; else cout << "Keys are not present "; lca = findLCA(root, 4, 10); if (lca != NULL) cout << "\nLCA(4, 10) = " << lca->key; else cout << "\nKeys are not present "; return 0; }

## Java

// Java implementation to find lowest common ancestor of // n1 and n2 using one traversal of binary tree // It also handles cases even when n1 and n2 are not there in Tree /* Class containing left and right child of current node and key */ class Node { int data; Node left, right; public Node(int item) { data = item; left = right = null; } } public class BinaryTree { // Root of the Binary Tree Node root; static boolean v1 = false, v2 = false; // This function returns pointer to LCA of two given // values n1 and n2. // v1 is set as true by this function if n1 is found // v2 is set as true by this function if n2 is found Node findLCAUtil(Node node, int n1, int n2) { // Base case if (node == null) return null; // If either n1 or n2 matches with root's key, report the presence // by setting v1 or v2 as true and return root (Note that if a key // is ancestor of other, then the ancestor key becomes LCA) if (node.data == n1) { v1 = true; return node; } if (node.data == n2) { v2 = true; return node; } // Look for keys in left and right subtrees Node left_lca = findLCAUtil(node.left, n1, n2); Node right_lca = findLCAUtil(node.right, n1, n2); // If both of the above calls return Non-NULL, then one key // is present in once subtree and other is present in other, // So this node is the LCA if (left_lca != null && right_lca != null) return node; // Otherwise check if left subtree or right subtree is LCA return (left_lca != null) ? left_lca : right_lca; } // Finds lca of n1 and n2 under the subtree rooted with 'node' Node findLCA(int n1, int n2) { // Initialize n1 and n2 as not visited v1 = false; v2 = false; // Find lca of n1 and n2 using the technique discussed above Node lca = findLCAUtil(root, n1, n2); // Return LCA only if both n1 and n2 are present in tree if (v1 && v2) return lca; // Else return NULL return null; } /* Driver program to test above functions */ public static void main(String args[]) { BinaryTree tree = new BinaryTree(); tree.root = new Node(1); tree.root.left = new Node(2); tree.root.right = new Node(3); tree.root.left.left = new Node(4); tree.root.left.right = new Node(5); tree.root.right.left = new Node(6); tree.root.right.right = new Node(7); Node lca = tree.findLCA(4, 5); if (lca != null) System.out.println("LCA(4, 5) = " + lca.data); else System.out.println("Keys are not present"); lca = tree.findLCA(4, 10); if (lca != null) System.out.println("LCA(4, 10) = " + lca.data); else System.out.println("Keys are not present"); } }

## Python

""" Program to find LCA of n1 and n2 using one traversal of Binary tree It handles all cases even when n1 or n2 is not there in tree """ # A binary tree node class Node: # Constructor to create a new node def __init__(self, key): self.key = key self.left = None self.right = None # This function retturn pointer to LCA of two given values # n1 and n2 # v1 is set as true by this function if n1 is found # v2 is set as true by this function if n2 is found def findLCAUtil(root, n1, n2, v): # Base Case if root is None: return None # IF either n1 or n2 matches ith root's key, report # the presence by setting v1 or v2 as true and return # root (Note that if a key is ancestor of other, then # the ancestor key becomes LCA) if root.key == n1 : v[0] = True return root if root.key == n2: v[1] = True return root # Look for keys in left and right subtree left_lca = findLCAUtil(root.left, n1, n2, v) right_lca = findLCAUtil(root.right, n1, n2, v) # If both of the above calls return Non-NULL, then one key # is present in once subtree and other is present in other, # So this node is the LCA if left_lca and right_lca: return root # Otherwise check if left subtree or right subtree is LCA return left_lca if left_lca is not None else right_lca def find(root, k): # Base Case if root is None: return False # If key is present at root, or if left subtree or right # subtree , return true if (root.key == k or find(root.left, k) or find(root.right, k)): return True # Else return false return False # This function returns LCA of n1 and n2 onlue if both # n1 and n2 are present in tree, otherwise returns None def findLCA(root, n1, n2): # Initialize n1 and n2 as not visited v = [False, False] # Find lac of n1 and n2 using the technique discussed above lca = findLCAUtil(root, n1, n2, v) # Returns LCA only if both n1 and n2 are present in tree if (v[0] and v[1] or v[0] and find(lca, n2) or v[1] and find(lca, n1)): return lca # Else return None return None # Driver program to test above function root = Node(1) root.left = Node(2) root.right = Node(3) root.left.left = Node(4) root.left.right = Node(5) root.right.left = Node(6) root.right.right = Node(7) lca = findLCA(root, 4, 5) if lca is not None: print "LCA(4, 5) = ", lca.key else : print "Keys are not present" lca = findLCA(root, 4, 10) if lca is not None: print "LCA(4,10) = ", lca.key else: print "Keys are not present" # This code is contributed by Nikhil Kumar Singh(nickzuck_007)

Output:

LCA(4, 5) = 2 Keys are not present

Thanks to Dhruv for suggesting this extended solution.

You may like to see below articles as well :

Lowest Common Ancestor in a Binary Search Tree.

Find LCA in Binary Tree using RMQ

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