Remove edges connected to a node such that the three given nodes are in different trees

Given a binary tree and 3 nodes a, b and c, the task is to find a node in the tree such that after removing all the edge connected to that node, a, b and c are in three different trees.

Given below is a tree with input nodes as c, j and o.
Example Tree

In the above tree, if node i gets disconnected from the tree, then the given nodes c, j, and o will be in three different trees which have been shown below.

Tree with edges removed from the selected node

A simple approach is to find LCA of all possible pairs of nodes given.
Let,

  • lca of ( a, b) = x
  • lca of (b, c) = y
  • lca of (c, a) = z

In any case, either of (x, y), (y, z), (z, x) or (x, y, z) will always be the same. In the first three cases, return the node which is not the same. In the last case returning any node of x, y or z will give the answer.

Below is the implementation of the above approach:

// C++ program for disconnecting a
// node to result in three different tree
#include <bits/stdc++.h>
using namespace std;
  
// node class
struct Node {
    int key;
    struct Node *left, *right;
};
Node* newNode(int key)
{
    Node* temp = new Node;
    temp->key = key;
    temp->left = temp->right = NULL;
    return (temp);
}
  
// LCA function taken from the above link mentioned
// This function returns a 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;
}
  
// the function assumes a, b, c are present in the tree
// and returns a node disconnecting which
// results in all three nodes in different trees
Node* findNode(Node* root, int a, int b, int c)
{
    // lca of a, b
    Node* x = findLCA(root, a, b);
  
    // lca of b, c
    Node* y = findLCA(root, b, c);
  
    // lca of c, a
    Node* z = findLCA(root, c, a);
  
    if (x->key == y->key)
        return z;
    else if (x->key == z->key)
        return y;
    else
        return x;
}
  
// Driver Code
int main()
{
    // Declare tree
    // Insert elements in the tree
    Node* root = newNode(1);
  
    root->left = newNode(2);
    root->right = newNode(3);
  
    root->left->left = newNode(4);
    root->left->right = newNode(5);
  
    root->left->left->left = newNode(8);
    root->left->left->right = newNode(9);
  
    root->left->right->left = newNode(10);
    root->left->right->right = newNode(11);
  
    root->right->left = newNode(6);
    root->right->right = newNode(7);
    root->right->left->left = newNode(12);
    root->right->left->right = newNode(13);
    root->right->right->left = newNode(14);
    root->right->right->right = newNode(15);
  
    /*
            1
        /     \
       2       3
     /  \     /  \
    4   5     6    7
   /\  / \   / \  / \
  8 9 10 11 12 13 14 15     
                           */
  
    // update all the suitable_children
    // keys of all the nodes in O( N )
  
    cout << "Disconnect node "
         << findNode(root, 5, 6, 15)->key
         << " from the tree";
  
    return 0;
}

Output:

Disconnect node 3 from the tree


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