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Check if Tree is Isomorphic

In this article, we are going to see how to check a tree whether isomorphic or not? This problem has been featured in interview coding round of Microsoft, Amazon.
Submitted by Radib Kar, on February 04, 2019

Problem statement:

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.

Example1:

 Isomorphic tree example 1

These two trees are isomorphic

Swap left child & right child of 1

 Isomorphic tree example 2

Swap left & right child of 5

 Isomorphic tree example 3

Example 2:

 Isomorphic tree example 4

Solution:

The conditions which needed to be satisfied are:

  1. Empty trees are isomorphic
  2. Roots must be the same
  3. Either left subtree & right subtree of one must be same with the same of other's, or left subtree of one must been same with right subtree of other's & right subtree of one must same with left subtree of other's.

Pre-requisite:

Two Input binary trees (their roots actually), i.e., root1, root2

FUNCTION isIsomorphic(Node *root1,Node *root2)
    1.  Both are empty then it's isomorphic. //condition-1
	    IF (!root1 && !root2)
	        return true;
	    If particularly exactly one of them is empty, 
        then they can't be isomorphic.//extension of condition-1
	    IF(!root1 || !root2)
	        return false;
    2.  If root value are different, they can't be isomorphic//condition-2
	    IF(root1->data!=root2->data)
	        return false;
    3.  Check condition-3
        Recursively checking subtrees
        return ( (  isIsomorphic(root1->left,root2->left) && 
                    isIsomorphic(root1->right,root2->right) ) || 
                   (isIsomorphic(root1->right,root2->left) &&
                    isIsomorphic(root1->left,root2->right)));
END FUNCTION

C++ implementation

#include <bits/stdc++.h>
using namespace std;

// TreeNode node type
class TreeNode{
	public:             
	int val;           //value
	TreeNode *left;    //pointer to left child
	TreeNode *right;   //pointer to right child
};

// creating new node
TreeNode* newnode(int data)  
{ 
	TreeNode* node = (TreeNode*)malloc(sizeof(TreeNode)); 
	node->val = data; 
	node->left = NULL; 
	node->right = NULL; 

	return(node); 
}

// function to check isomorphic trees
bool isIsomorphic(TreeNode *root1,TreeNode *root2)
{
	if(!root1 && !root2)
	return true;

	if(!root1 || !root2)
	return false;

	if(root1->val!=root2->val)
	return false;

	return ( (isIsomorphic(root1->left,root2->left) && 
			isIsomorphic(root1->right,root2->right) )|| 
			(isIsomorphic(root1->right,root2->left) && 
			isIsomorphic(root1->left,root2->right)));
}

int main(){
	cout<<"tree is built as per example-1\n";

	TreeNode *root1=newnode(1); 
	root1->left= newnode(5); 
	root1->right= newnode(2); 
	root1->right->right=newnode(3);
	root1->left->left=newnode(8); 
	root1->left->right=newnode(4);

	TreeNode *root2=newnode(1); 
	root2->left= newnode(2); 
	root2->right= newnode(5); 
	root2->right->right=newnode(8);
	root2->right->left=newnode(4);
	root2->left->right=newnode(3);

	if(isIsomorphic(root1,root2))
		cout<<"They are isomorphic tree\n";
	else
		cout<<"They are not isomorphic tree\n";

	cout<<"tree is built as per example-2\n";

	TreeNode *root3=newnode(1); 
	root3->left= newnode(9); 
	root3->right= newnode(2); 
	root3->right->right=newnode(3);
	root3->left->left=newnode(8); 
	root3->left->right=newnode(4);

	if(isIsomorphic(root1,root3))
		cout<<"They are isomorphic tree\n";
	else
		cout<<"They are not isomorphic tree\n";

	return 0;
}

Output

tree is built as per example-1
They are isomorphic tree
tree is built as per example-2
They are not isomorphic tree

Explanation with example

Let's check the example-1

Nodes are represented by their respective values for better understanding

In the main we call isIsomorphic(root1,root2) //isIsomorphic(1,1)
------------------------------------------------

isIsomorphic(1,1)
root1 is not NULL
root2 is not NULL
root1->data==root2->data
thus it returns
((isIsomorphic(root1->left,root2->left) && isIsomorphic(root1->right,root2->right)) || 
(isIsomorphic(root1->right,root2->left) && isIsomorphic(root1->left,root2->right)));
i.e.,
((isIsomorphic(5,2) && isIsomorphic(2,5)) || 
(isIsomorphic(2,2) && isIsomorphic(5,5)));

Thus call to isIsomorphic(5,2), isIsomorphic(2,5), 
isIsomorphic(2,2), isIsomorphic(5,5)
------------------------------------------------

isIsomorphic(5,2)
root1 is not NULL
root2 is not NULL
root1->data!=root2->data
thus it returns FALSE
------------------------------------------------

isIsomorphic(2,5)
root1 is not NULL
root2 is not NULL
root1->data!=root2->data
thus it returns FALSE
------------------------------------------------

isIsomorphic(2,2)
root1 is not NULL
root2 is not NULL
root1->data==root2->data
thus it returns 
((isIsomorphic(root1->left,root2->left) && isIsomorphic(root1->right,root2->right) ) || 
(isIsomorphic(root1->right,root2->left) && isIsomorphic(root1->left,root2->right)));
i.e.,
((isIsomorphic(NULL,NULL) && isIsomorphic(3,3)) || 
(isIsomorphic(3,NULL) && isIsomorphic(NULL,3)));

Thus call to isIsomorphic(NULL,NULL), isIsomorphic(3,3), 
isIsomorphic(3,NULL), isIsomorphic(NULL,3)
------------------------------------------------

isIsomorphic(NULL,NULL)
root1 is NULL
root2 is NULL
thus it return TRUE
------------------------------------------------

isIsomorphic(3,3)
root1 is not NULL
root2 is not NULL
root1->data==root2->data
thus it returns 
((isIsomorphic(root1->left,root2->left) && isIsomorphic(root1->right,root2->right) ) || 
(isIsomorphic(root1->right,root2->left) && isIsomorphic(root1->left,root2->right)));
i.e.,
((isIsomorphic(NULL,NULL) && (isIsomorphic(NULL,NULL) || 
(isIsomorphic(NULL,NULL) && (isIsomorphic(NULL,NULL)));

Thus call to isIsomorphic(NULL,NULL), (isIsomorphic(NULL,NULL),
(isIsomorphic(NULL,NULL), (isIsomorphic(NULL,NULL))

All (isIsomorphic(NULL,NULL) returns TRUE
Thus, isIsomorphic(3,3) returns TRUE
------------------------------------------------

isIsomorphic(3,NULL)
exactly one root is empty
Thus it returns FALSE
------------------------------------------------

isIsomorphic(NULL,3)
exactly one root is empty
Thus it returns FALSE

------------------------------------------------

Thus ,
isIsomorphic(2,2)
=((isIsomorphic(NULL,NULL) && isIsomorphic(3,3) ) || 
(isIsomorphic(3,NULL) && isIsomorphic(NULL,3)))
=(TRUE && TRUE)|| (FALSE && FALSE)
=TRUE && FLASE
=TRUE

Same way, we can check that isIsomorphic(5,5) returns TRUE
------------------------------------------------

At main:
isIsomorphic(1,1)
=((isIsomorphic(5,2) && isIsomorphic(2,5)) || 
(isIsomorphic(2,2) && isIsomorphic(5,5)))
=((FALSE && FALSE)||(TRUE && TRUE)
=(FALSE && TRUE)
TRUE

Thus these two trees are isomorphic.

You can check the second example same way & can find returning FALSE






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