Tree Isomorphism Problem

Write a function to detect if two trees are isomorphic. Two trees are called isomorphic if one of them can be obtained from other by a series of flips, i.e. by swapping left and right children of a number of nodes. Any number of nodes at any level can have their children swapped. Two empty trees are isomorphic.

For example, following two trees are isomorphic with following sub-trees flipped: 2 and 3, NULL and 6, 7 and 8.

We simultaneously traverse both trees. Let the current internal nodes of two trees being traversed be n1 and n2 respectively. There are following two conditions for subtrees rooted with n1 and n2 to be isomorphic.
1) Data of n1 and n2 is same.
2) One of the following two is true for children of n1 and n2
……a) Left child of n1 is isomorphic to left child of n2 and right child of n1 is isomorphic to right child of n2.
……b) Left child of n1 is isomorphic to right child of n2 and right child of n1 is isomorphic to left child of n2.

// A C++ program to check if two given trees are isomorphic
#include <iostream>
using namespace std;
/* A binary tree node has data, pointer to left and right children */
struct node
    int data;
    struct node* left;
    struct node* right;
/* Given a binary tree, print its nodes in reverse level order */
bool isIsomorphic(node* n1, node *n2)
 // Both roots are NULL, trees isomorphic by definition
 if (n1 == NULL && n2 == NULL)
    return true;
 // Exactly one of the n1 and n2 is NULL, trees not isomorphic
 if (n1 == NULL || n2 == NULL)
    return false;
 if (n1->data != n2->data)
    return false;
 // There are two possible cases for n1 and n2 to be isomorphic
 // Case 1: The subtrees rooted at these nodes have NOT been "Flipped".
 // Both of these subtrees have to be isomorphic, hence the &&
 // Case 2: The subtrees rooted at these nodes have been "Flipped"
 (isIsomorphic(n1->left,n2->left) && isIsomorphic(n1->right,n2->right))||
 (isIsomorphic(n1->left,n2->right) && isIsomorphic(n1->right,n2->left));
/* Helper function that allocates a new node with the
   given data and NULL left and right pointers. */
node* newNode(int data)
    node* temp = new node;
    temp->data = data;
    temp->left = NULL;
    temp->right = NULL;
    return (temp);
/* Driver program to test above functions*/
int main()
    // Let us create trees shown in above diagram
    struct node *n1 = newNode(1);
    n1->left        = newNode(2);
    n1->right       = newNode(3);
    n1->left->left  = newNode(4);
    n1->left->right = newNode(5);
    n1->right->left  = newNode(6);
    n1->left->right->left = newNode(7);
    n1->left->right->right = newNode(8);
    struct node *n2 = newNode(1);
    n2->left         = newNode(3);
    n2->right        = newNode(2);
    n2->right->left   = newNode(4);
    n2->right->right   = newNode(5);
    n2->left->right   = newNode(6);
    n2->right->right->left = newNode(8);
    n2->right->right->right = newNode(7);
    if (isIsomorphic(n1, n2) == true)
       cout << "Yes";
      cout << "No";
    return 0;



Time Complexity: The above solution does a traversal of both trees. So time complexity is O(m + n) where m and n are number of nodes in given trees.

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