Maximum XOR path of a Binary Tree

Given a Binary Tree, the task is to find the maximum of all the XOR value of all the nodes in the path from the root to leaf.

Examples:

Input: 
       2
      / \
     1   4
    / \   
   10  8   
Output: 11
Explanation:
All the paths are: 
2-1-10 XOR-VALUE = 9
2-1-8 XOR-VALUE = 11
2-4 XOR-VALUE = 6

Input: 
        2
      /   \
     1     4
    / \   / \
   10  8 5  10
Output: 12

Approach:

  1. To solve the question mentioned above we have to traverse the tree recursively using pre-order traversal. For each node keep calculating the XOR of the path from root till the current node.

    XOR of current node’s path = (XOR of the path till the parent) ^ (current node value)



  2. If the node is a leaf node that is left and the right child for the current nodes are NULL then we compute the max-Xor, as

    max-Xor = max(max-Xor, cur-Xor).

Below is the implementation of the above approach:

C++

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// C++ program to compute the
// Max-Xor value of path from
// the root to leaf of a Binary tree
  
#include <bits/stdc++.h>
using namespace std;
  
// Binary tree node
struct Node {
    int data;
  
    struct Node *left, *right;
};
  
// Function to create a new node
struct Node* newNode(int data)
{
    struct Node* newNode = new Node;
  
    newNode->data = data;
  
    newNode->left
        = newNode->right = NULL;
  
    return (newNode);
}
  
// Function calculate the
// value of max-xor
void Solve(Node* root, int xr,
           int& max_xor)
{
  
    // Updating the xor value
    // with the xor of the
    // path from root to
    // the node
    xr = xr ^ root->data;
  
    // Check if node is leaf node
    if (root->left == NULL
        && root->right == NULL) {
  
        max_xor = max(max_xor, xr);
        return;
    }
  
    // Check if the left
    // node exist in the tree
    if (root->left != NULL) {
        Solve(root->left, xr,
              max_xor);
    }
  
    // Check if the right node
    // exist in the tree
    if (root->right != NULL) {
        Solve(root->right, xr,
              max_xor);
    }
  
    return;
}
  
// Function to find the
// required count
int findMaxXor(Node* root)
{
  
    int xr = 0, max_xor = 0;
  
    // Recursively traverse
    // the tree and compute
    // the max_xor
    Solve(root, xr, max_xor);
  
    // Return the result
    return max_xor;
}
  
// Driver code
int main(void)
{
    // Create the binary tree
    struct Node* root = newNode(2);
    root->left = newNode(1);
    root->right = newNode(4);
    root->left->left = newNode(10);
    root->left->right = newNode(8);
    root->right->left = newNode(5);
    root->right->right = newNode(10);
  
    cout << findMaxXor(root);
  
    return 0;
}

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Python3

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# Python3 program to compute the 
# Max-Xor value of path from 
# the root to leaf of a Binary tree 
  
# Binary tree node
class Node:
      
    # Function to create a new node
    def __init__(self, data):
          
        self.data = data
        self.left = None
        self.right = None
  
# Function calculate the 
# value of max-xor
def Solve(root, xr, max_xor):
      
    # Updating the xor value 
    # with the xor of the 
    # path from root to 
    # the node
    xr = xr ^ root.data
      
    # Check if node is leaf node
    if (root.left == None and 
        root.right == None):
        max_xor[0] = max(max_xor[0], xr)
      
    # Check if the left 
    # node exist in the tree
    if root.left != None:
        Solve(root.left, xr, max_xor)
      
    # Check if the right node 
    # exist in the tree 
    if root.right != None:
        Solve(root.right, xr, max_xor)
          
    return
  
# Function to find the 
# required count 
def findMaxXor(root):
      
    xr, max_xor = 0, [0]
      
    # Recursively traverse 
    # the tree and compute 
    # the max_xor 
    Solve(root, xr, max_xor)
      
    # Return the result
    return max_xor[0]
  
# Driver code
  
# Create the binary tree
root = Node(2)
root.left = Node(1)
root.right = Node(4
root.left.left = Node(10
root.left.right = Node(8
root.right.left = Node(5
root.right.right = Node(10
  
print(findMaxXor(root))
  
# This code is contributed by Shivam Singh

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Output:

12

Time Complexity: We are iterating over each node only once, therefore it will take O(N) time where N is the number of nodes in the Binary tree.
Auxiliary Space Complexity: The Auxiliary Space complexity will be O(1), as there is no extra space used

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