Find mirror of a given node in Binary tree

Given a Binary tree, the problem is to find the mirror of a given node. The mirror of a node is a node which exists at the mirror position of the node in opposite subtree at the root.

Examples:

In above tree-
Node 2 and 3 are mirror nodes
Node 4 and 6 are mirror nodes. 

We can have a recursive solution for finding mirror nodes. The algorithm is following –  

1) Start from the root of the tree and recur 
   nodes from both subtree simultaneously 
   using two pointers for left and right nodes.
2) First recur all the external nodes and 
   store returned value in mirror variable.
3) If current node value is equal to target node, 
   return the value of opposite pointer else 
   repeat step 2.
4) If no external node is left and mirror is 
   none, recur internal nodes.

Implementation:

C++

// C++ program to find the mirror Node
// in Binary tree
#include <bits/stdc++.h>

using namespace std;

/* A binary tree Node has data,
pointer to left child and
a pointer to right child */
struct Node
{
    int key;
    struct Node* left, *right;
};

// create new Node and initialize it
struct Node* newNode(int key)
{
    struct Node* n = (struct Node*)
                      malloc(sizeof(struct Node*));
    if (n != NULL)
    {
        n->key = key;
        n->left = NULL;
        n->right = NULL;
        return n;
    }
    else
    {
        cout << "Memory allocation failed!"
             << endl;
        exit(1);
    }
}

// recursive function to find mirror of Node
int findMirrorRec(int target, struct Node* left,
                              struct Node* right)
{
    /* if any of the Node is none then Node itself
    and descendant have no mirror, so return
    none, no need to further explore! */
    if (left == NULL || right == NULL)
        return 0;

    /* if left Node is target Node, then return
    right's key (that is mirror) and vice
    versa */
    if (left->key == target)
        return right->key;

    if (right->key == target)
        return left->key;

    // first recur external Nodes
    int mirror_val = findMirrorRec(target,
                                   left->left,
                                   right->right);
    if (mirror_val)
        return mirror_val;

    // if no mirror found, recur internal Nodes
    findMirrorRec(target, left->right, right->left);
}

// interface for mirror search
int findMirror(struct Node* root, int target)
{
    if (root == NULL)
        return 0;
    if (root->key == target)
        return target;
    return findMirrorRec(target, root->left,
                                 root->right);
}

// Driver Code
int main()
{
    struct Node* root = newNode(1);
    root-> left = newNode(2);
    root->left->left = newNode(4);
    root->left->left->right    = newNode(7);
    root->right    = newNode(3);
    root->right->left = newNode(5);
    root->right->right = newNode(6);
    root->right->left->left    = newNode(8);
    root->right->left->right = newNode(9);

    // target Node whose mirror have to be searched
    int target = root->left->left->key;

    int mirror = findMirror(root, target);

    if (mirror)
        cout << "Mirror of Node " << target
             << " is Node " << mirror << endl;
    else
        cout << "Mirror of Node " << target
             << " is NULL! " << endl;
}

// This code is contributed by SHUBHAMSINGH10

C

// C program to find the mirror Node in Binary tree
#include <stdio.h>
#include <stdlib.h>

/* A binary tree Node has data, pointer to left child
  and a pointer to right child */
struct Node
{
    int key;
    struct Node* left, *right;
};

// create new Node and initialize it
struct Node* newNode(int key)
{
    struct Node* n = (struct Node*)
                     malloc(sizeof(struct Node*));
    if (n != NULL)
    {
        n->key = key;
        n->left = NULL;
        n->right = NULL;
        return n;
    }
    else
    {
        printf("Memory allocation failed!");
        exit(1);
    }
}

// recursive function to find mirror of Node
int findMirrorRec(int target, struct Node* left,
                              struct Node* right)
{
    /* if any of the Node is none then Node itself
       and descendant have no mirror, so return
       none, no need to further explore! */
    if (left==NULL || right==NULL)
        return 0;

    /* if left Node is target Node, then return
       right's key (that is mirror) and vice
       versa */
    if (left->key == target)
        return right->key;

    if (right->key == target)
        return left->key;

    // first recur external Nodes
    int mirror_val = findMirrorRec(target,
                                     left->left,
                                     right->right);
    if (mirror_val)
        return mirror_val;

    // if no mirror found, recur internal Nodes
    findMirrorRec(target, left->right, right->left);
}

// interface for mirror search
int findMirror(struct Node* root, int target)
{
    if (root == NULL)
        return 0;
    if (root->key == target)
        return target;
    return findMirrorRec(target, root->left, root->right);
}

// Driver
int main()
{
    struct Node* root           = newNode(1);
    root-> left                 = newNode(2);
    root->left->left            = newNode(4);
    root->left->left->right     = newNode(7);
    root->right                 = newNode(3);
    root->right->left           = newNode(5);
    root->right->right          = newNode(6);
    root->right->left->left     = newNode(8);
    root->right->left->right    = newNode(9);

    // target Node whose mirror have to be searched
    int target = root->left->left->key;

    int mirror = findMirror(root, target);

    if (mirror)
        printf("Mirror of Node %d is Node %d\n",
                                    target, mirror);
    else
        printf("Mirror of Node %d is NULL!\n", target);
}

Java

// Java program to find the mirror Node in Binary tree
class GfG {

/* A binary tree Node has data, pointer to left child
and a pointer to right child */
static class Node
{
    int key;
    Node left, right;
}

// create new Node and initialize it
static Node newNode(int key)
{
    Node n = new Node();

        n.key = key;
        n.left = null;
        n.right = null;
        return n;
}

// recursive function to find mirror of Node
static int findMirrorRec(int target, Node left, Node right)
{
    /* if any of the Node is none then Node itself
    and descendant have no mirror, so return
    none, no need to further explore! */
    if (left==null || right==null)
        return 0;

    /* if left Node is target Node, then return
    right's key (that is mirror) and vice
    versa */
    if (left.key == target)
        return right.key;

    if (right.key == target)
        return left.key;

    // first recur external Nodes
    int mirror_val = findMirrorRec(target, left.left, right.right);
    if (mirror_val != 0)
        return mirror_val;

    // if no mirror found, recur internal Nodes
    return findMirrorRec(target, left.right, right.left);
}

// interface for mirror search
static int findMirror(Node root, int target)
{
    if (root == null)
        return 0;
    if (root.key == target)
        return target;
    return findMirrorRec(target, root.left, root.right);
}

// Driver
public static void main(String[] args)
{
    Node root         = newNode(1);
    root.left                 = newNode(2);
    root.left.left         = newNode(4);
    root.left.left.right     = newNode(7);
    root.right                 = newNode(3);
    root.right.left         = newNode(5);
    root.right.right         = newNode(6);
    root.right.left.left     = newNode(8);
    root.right.left.right = newNode(9);

    // target Node whose mirror have to be searched
    int target = root.left.left.key;

    int mirror = findMirror(root, target);

    if (mirror != 0)
        System.out.println("Mirror of Node " + target + " is Node " + mirror);
    else
        System.out.println("Mirror of Node " + target + " is null ");
}
}

Python3

# Python3 program to find the mirror node in
# Binary tree

class Node:
    '''A binary tree node has data, reference to left child
         and a reference to right child '''

    def __init__(self, key, lchild=None, rchild=None):
        self.key = key
        self.lchild = None
        self.rchild = None


# recursive function to find mirror
def findMirrorRec(target, left, right):

    # If any of the node is none then node itself
    # and descendant have no mirror, so return
    # none, no need to further explore!
    if left == None or right == None:
        return None

    # if left node is target node, then return
    # right's key (that is mirror) and vice versa
    if left.key == target:
        return right.key
    if right.key == target:
        return left.key

    # first recur external nodes
    mirror_val = findMirrorRec(target, left.lchild, right.rchild)
    if mirror_val != None:
        return mirror_val

    # if no mirror found, recur internal nodes
    findMirrorRec(target, left.rchild, right.lchild)

# interface for mirror search
def findMirror(root, target):
    if root == None:
        return None

    if root.key == target:
        return target

    return findMirrorRec(target, root.lchild, root.rchild)

# Driver
def main():
    root = Node(1)
    n1 = Node(2)
    n2 = Node(3)
    root.lchild = n1
    root.rchild = n2
    n3 = Node(4)
    n4 = Node(5)
    n5 = Node(6)
    n1.lchild = n3
    n2.lchild = n4
    n2.rchild = n5
    n6 = Node(7)
    n7 = Node(8)
    n8 = Node(9)
    n3.rchild = n6
    n4.lchild = n7
    n4.rchild = n8

    # target node whose mirror have to be searched
    target = n3.key

    mirror = findMirror(root, target)
    print("Mirror of node {} is node {}".format(target, mirror))

if __name__ == '__main__':
    main()

C#

// C# program to find the
// mirror Node in Binary tree
using System;

class GfG
{

    /* A binary tree Node has data,
        pointer to left child and a
        pointer to right child */
    class Node
    {
        public int key;
        public Node left, right;
    }

    // create new Node and initialize it
    static Node newNode(int key)
    {
        Node n = new Node();

            n.key = key;
            n.left = null;
            n.right = null;
            return n;
    }

    // recursive function to find mirror of Node
    static int findMirrorRec(int target, Node left,
                                        Node right)
    {
        /* if any of the Node is none then Node itself
        and descendant have no mirror, so return
        none, no need to further explore! */
        if (left==null || right==null)
            return 0;

        /* if left Node is target Node, then return
        right's key (that is mirror) and vice
        versa */
        if (left.key == target)
            return right.key;

        if (right.key == target)
            return left.key;

        // first recur external Nodes
        int mirror_val = findMirrorRec(target,
                        left.left, right.right);
        if (mirror_val != 0)
            return mirror_val;

        // if no mirror found, recur internal Nodes
        return findMirrorRec(target,
                left.right, right.left);
    }

    // interface for mirror search
    static int findMirror(Node root, int target)
    {
        if (root == null)
            return 0;
        if (root.key == target)
            return target;
        return findMirrorRec(target,
                root.left, root.right);
    }

    // Driver code
    public static void Main(String[] args)
    {
        Node root = newNode(1);
        root.left = newNode(2);
        root.left.left = newNode(4);
        root.left.left.right = newNode(7);
        root.right = newNode(3);
        root.right.left    = newNode(5);
        root.right.right = newNode(6);
        root.right.left.left = newNode(8);
        root.right.left.right = newNode(9);

        // target Node whose mirror have to be searched
        int target = root.left.left.key;

        int mirror = findMirror(root, target);

        if (mirror != 0)
            Console.WriteLine("Mirror of Node " +
                                target + " is Node " +
                                            mirror);
        else
            Console.WriteLine("Mirror of Node " + target +
                                            " is null ");
    }
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// Javascript program to find the mirror
// Node in Binary tree

/* A binary tree Node has data, pointer
to left child and a pointer to right child */
class Node
{

    // Create new Node and initialize it
    constructor(key)
    {
        this.key = key;
        this.left = this.right = null;
    }
}

// Recursive function to find mirror of Node
function findMirrorRec(target, left, right)
{

    /* If any of the Node is none then Node itself
    and descendant have no mirror, so return
    none, no need to further explore! */
    if (left == null || right == null)
        return 0;

    /* If left Node is target Node, then return
    right's key (that is mirror) and vice
    versa */
    if (left.key == target)
        return right.key;

    if (right.key == target)
        return left.key;

    // First recur external Nodes
    let mirror_val = findMirrorRec(
        target, left.left, right.right);
    if (mirror_val != 0)
        return mirror_val;

    // If no mirror found, recur internal Nodes
    return findMirrorRec(target, left.right,
                                 right.left);
}

// Interface for mirror search
function findMirror(root, target)
{
    if (root == null)
        return 0;
    if (root.key == target)
        return target;

    return findMirrorRec(target, root.left,
                                 root.right);
}

// Driver code
let root = new Node(1);
root.left = new Node(2);
root.left.left = new Node(4);
root.left.left.right = new Node(7);
root.right = new Node(3);
root.right.left = new Node(5);
root.right.right = new Node(6);
root.right.left.left = new Node(8);
root.right.left.right = new Node(9);

// Target Node whose mirror have to be searched
let target = root.left.left.key;

let mirror = findMirror(root, target);

if (mirror != 0)
    document.write("Mirror of Node " + target +
                   " is Node " + mirror + "<br>");
else
    document.write("Mirror of Node " + target +
                   " is null " + "<br>");

// This code is contributed by rag2127

</script>

Output

Mirror of Node 4 is Node 6

Time Complexity:

Space Complexity :-The space complexity is O(H), where H is the height of the tree, due to the recursive call stack.

Another Approach:-

• So, the thought process behind this problem is that as we know that the mirror image will be in the same of the node.
• So, it is clear that we have to look into the same level.
• But also we have to find the position of the image which is mirror of the given node because all node in a level can not be the mirror node of a given node.
• So to do so we have to first find out the level as well as the position of the given node, So that we can find the mirror image.
• In initial moment we will take the position of the given node and level of the node as -1
• After this we will start traversing the tree using DFS algorithm and increase the level in each call as we are going down and set the position according to the call like if we are going left then we will decrease the position else we will increase the position by 1.
• As soon as we got the node with given target value we will set the position and level of the target node which was previously -1 to the current position and level.
• After this we have to check if we got the level and position of the target node and we are at the same level then we have to find out the mirror position and that we can identify by doing abs of the current position and target position because the position of any one or both can be negative.
• If we found the same abs(position) at same level then this is the mirror node, store this in your answer.
• Below is the implementation of this approach:-

Implementation:-

C++

// C++ program to find the mirror Node
// in Binary tree
#include <bits/stdc++.h>

using namespace std;

/* A binary tree Node has data,
pointer to left child and
a pointer to right child */
struct Node
{
    int key;
    struct Node* left, *right;
};

// create new Node and initialize it
struct Node* newNode(int key)
{
    struct Node* n = (struct Node*)
                      malloc(sizeof(struct Node*));
    if (n != NULL)
    {
        n->key = key;
        n->left = NULL;
        n->right = NULL;
        return n;
    }
    else
    {
        cout << "Memory allocation failed!"
             << endl;
        exit(1);
    }
}
void findMirror(Node* root, int target, int &target_level, int &target_position,
                int level, int position,int &mirror)
{
      if(!root)return;

      //if node is target node
      if(root->key==target){
      target_level=level;
      target_position = position;
      return;
    }

      //checking if target has found then check for mirro
      if(target_level!=-1 and level==target_level and (abs(position)==abs(target_position))){
      mirror=root->key;
      return;
    }

      //going left side of the node
      findMirror(root->left,target,target_level,target_position,level+1,position-1,mirror);

      //going right side of the node
      findMirror(root->right,target,target_level,target_position,level+1,position+1,mirror);
}
// Driver Code
int main()
{
    struct Node* root = newNode(1);
    root-> left = newNode(2);
    root->left->left = newNode(4);
    root->left->left->right    = newNode(7);
    root->right    = newNode(3);
    root->right->left = newNode(5);
    root->right->right = newNode(6);
    root->right->left->left    = newNode(8);
    root->right->left->right = newNode(9);

    // target Node whose mirror have to be searched
    int target = root->left->left->key;

    int mirror = 0;

      int target_level=-1,target_position=-1;

    findMirror(root, target,target_level,target_position,0,0,mirror);

    if (mirror!=0)
        cout << "Mirror of Node " << target
             << " is Node " << mirror << endl;
    else
        cout << "Mirror of Node " << target
             << " is NULL! " << endl;
}

// This code is contributed by shubhamrajput6156

Java

import java.util.*;

public class Gfg {

    /* A binary tree Node has data,
       pointer to left child and
       a pointer to right child */
    static class Node {
        int key;
        Node left, right;
    };

    // create new Node and initialize it
    static Node newNode(int key) {
        Node n = new Node();
        n.key = key;
        n.left = null;
        n.right = null;
        return n;
    }

    static void findMirror(Node root, int target, int[] targetLevel, int[] targetPosition,
                            int level, int position, int[] mirror) {
        if (root == null) return;

        // if node is target node
        if (root.key == target) {
            targetLevel[0] = level;
            targetPosition[0] = position;
            return;
        }

        // checking if target has found then check for mirror
        if (targetLevel[0] != -1 && level == targetLevel[0] && Math.abs(position) == Math.abs(targetPosition[0])) {
            mirror[0] = root.key;
            return;
        }

        // going left side of the node
        findMirror(root.left, target, targetLevel, targetPosition, level + 1, position - 1, mirror);

        // going right side of the node
        findMirror(root.right, target, targetLevel, targetPosition, level + 1, position + 1, mirror);
    }

    // Driver Code
    public static void main(String[] args) {
        Node root = newNode(1);
        root.left = newNode(2);
        root.left.left = newNode(4);
        root.left.left.right = newNode(7);
        root.right = newNode(3);
        root.right.left = newNode(5);
        root.right.right = newNode(6);
        root.right.left.left = newNode(8);
        root.right.left.right = newNode(9);

        // target Node whose mirror have to be searched
        int target = root.left.left.key;

        int[] mirror = new int[1];
        int[] targetLevel = {-1};
        int[] targetPosition = {0};

        findMirror(root, target, targetLevel, targetPosition, 0, 0, mirror);

        if (mirror[0] != 0) {
            System.out.println("Mirror of Node " + target + " is Node " + mirror[0]);
        } else {
            System.out.println("Mirror of Node " + target + " is NULL!");
        }
    }
}

Python3

# Python program to find the mirror Node
# in Binary tree

# A binary tree Node has data,
# pointer to left child and
# a pointer to right child
class Node:
    def __init__(self, key):
        self.key = key
        self.left = None
        self.right = None

target_level = -1
target_position = -1
mirror = 0

def findMirror(root, target, level, position):
    global target_level, target_position, mirror

    if not root:
        return

    # If node is target node
    if root.key == target:
        target_level = level
        target_position = position
        return

    # Checking if target has found then check for mirror
    if target_level != -1 and level == target_level and abs(position) == abs(target_position):
        mirror = root.key
        return

    # Going left side of the node
    findMirror(root.left, target, level + 1, position - 1)

    # Going right side of the node
    findMirror(root.right, target, level + 1, position + 1)


# Driver Code
root = Node(1)
root.left = Node(2)
root.left.left = Node(4)
root.left.left.right = Node(7)
root.right = Node(3)
root.right.left = Node(5)
root.right.right = Node(6)
root.right.left.left = Node(8)
root.right.left.right = Node(9)

# Target Node whose mirror have to be searched
target = root.left.left.key

findMirror(root, target, 0, 0)

if mirror != 0:
    print(f"Mirror of Node {target} is Node {mirror}")
else:
    print(f"Mirror of Node {target} is NULL!")

Javascript

// JavaScript program to find the mirror Node
// in Binary tree

/* A binary tree Node has data,
pointer to left child and
a pointer to right child */
class Node {
    constructor(key) {
        this.key = key;
        this.left = null;
        this.right = null;
    }
}

let target_level = -1;
let target_position = -1;
let mirror = 0;
function findMirror(root, target, level, position) {
    if (!root) return;

    // If node is target node
    if (root.key == target) {
        target_level = level;
        target_position = position;
        return;
    }

    // Checking if target has found then check for mirror
    if (target_level != -1 && level == target_level && (Math.abs(position) ==   Math.abs(target_position))) {
        mirror = root.key;
        return;
    }

    // Going left side of the node
    findMirror(root.left, target, level + 1, position - 1);

    // Going right side of the node
    findMirror(root.right, target, level + 1, position + 1);
}

// Driver Code
let root = new Node(1);
root.left = new Node(2);
root.left.left = new Node(4);
root.left.left.right = new Node(7);
root.right = new Node(3);
root.right.left = new Node(5);
root.right.right = new Node(6);
root.right.left.left = new Node(8);
root.right.left.right = new Node(9);

// Target Node whose mirror have to be searched
let target = root.left.left.key;

findMirror(root, target, 0, 0);
if (mirror != 0) {
    console.log(`Mirror of Node ${target} is Node ${mirror}`);
} else {
    console.log(`Mirror of Node ${target} is NULL!`);
}

C#

using System;

public class Node
{
    public int key;
    public Node left, right;
}

public class BinaryTree
{
    public Node newNode(int key)
    {
        Node n = new Node();
        n.key = key;
        n.left = null;
        n.right = null;
        return n;
    }

    public void findMirror(Node root, int target, ref int target_level, ref int target_position,
                int level, int position, ref int mirror)
    {
        if (root == null) return;

        // if node is target node
        if (root.key == target)
        {
            target_level = level;
            target_position = position;
            return;
        }

        // checking if target has found then check for mirror
        if (target_level != -1 && level == target_level && (Math.Abs(position) == Math.Abs(target_position)))
        {
            mirror = root.key;
            return;
        }

        // going left side of the node
        findMirror(root.left, target, ref target_level, ref target_position, level + 1, position - 1, ref mirror);

        // going right side of the node
        findMirror(root.right, target, ref target_level, ref target_position, level + 1, position + 1, ref mirror);
    }

    public static void Main(string[] args)
    {
        BinaryTree tree = new BinaryTree();

        Node root = tree.newNode(1);
        root.left = tree.newNode(2);
        root.left.left = tree.newNode(4);
        root.left.left.right = tree.newNode(7);
        root.right = tree.newNode(3);
        root.right.left = tree.newNode(5);
        root.right.right = tree.newNode(6);
        root.right.left.left = tree.newNode(8);
        root.right.left.right = tree.newNode(9);

        // target Node whose mirror have to be searched
        int target = root.left.left.key;

        int mirror = 0;

        int target_level = -1, target_position = -1;

        tree.findMirror(root, target, ref target_level, ref target_position, 0, 0, ref mirror);

        if (mirror != 0)
            Console.WriteLine("Mirror of Node " + target + " is Node " + mirror);
        else
            Console.WriteLine("Mirror of Node " + target + " is NULL!");
    }
}

Output

Mirror of Node 4 is Node 6

Time Complexity:- O(N) where N is number of nodes in the tree

Auxiliary Space:- O(H) Recursive stack, Where H is height of tree.