C++ Program to Find the Lowest Common Ancestor of a Binary Search Tree

This is a C++ Program to find the Lowest Common Ancestor of a given tree.

Problem Description

We will be given a Binary Tree and we have to write a C++ program to find out the Lowest Common Ancestor of the two nodes of same tree taken as input from user.
Lowest Common Ancestor: In a given tree, the lowest common ancestor of two nodes node_1 and node_2 will be a node X such that node X will be the lowest node who has node_1 and node_2 as its descendants or children.

Expected Input and Output

Case 1. When both the nodes lie on same side of the root node and at same level:
For example :

If the input tree is 
             20
           /    \
          8     22
         /  \
        4   12
           /  \
          10  14
and the nodes are node_1 = 10, node_2 = 14,
then Output will be LCA = 12.

Case 2. When one of the nodes itself is a lowest common ancestor:
For example :

If the input tree is 
             20
           /    \
          8     22
         /  \
        4   12
           /  \
          10  14
and the nodes are node_1 = 14, node_2 = 8,
then Output will be LCA = 8.

Case 3. When the two nodes lie on different sides of root node:
For example :

If the input tree is 
             20
           /    \
          8     22
         /  \
        4   12
           /  \
          10  14
and the nodes are node_1 = 10, node_2 = 22,
then Output will be LCA = 20.

Problem Solution

1. First we need to look for node_1 and node_2 in given tree. If they lie on different sides of root node, then the root itself will be the LCA of node_1 and node_2.
2. If root is greater than node_1 and node_2 then their LCA will lie on left subtree.
3. If root is less than node_1 and node_2, their LCA will lie on right subtree.

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Program/Source Code

Here is source code of the C++ Program for finding the lowest common ancestor of of nodes in a given binary search tree. The program is successfully compiled and tested using Codeblocks gnu/gcc compiler on windows 10. The program output is also shown below.

```
/*
```

 * C++ Program to Find Lowest Common Ancestor in a Binary Search Tree

```
 */
```
```
#include<conio.h>
```
```
#include <stdio.h>
```
```
#include<iostream>
```
```
using namespace std;
```
```
struct node
```
```
{
```
```
    int data;
```
```
    struct node *left, *right;
```
```
};
```
```
/*
```
```
 * class declaration
```
```
 */
```
```
class BST
```
```
{
```
```
public:
```

    struct node *lca(struct node *root, int node_1, int node_2)

```
    {
```
```
        if (root != NULL)
```
```
        {
```

            if (root->data > node_1 && root->data > node_2)

```
            {
```

                return lca(root->left, node_1, node_2);

```
            }
```

            if (root->data < node_1 && root->data < node_2)

```
            {
```

                return lca(root->right, node_1, node_2);

```
            }
```
```
            return root;
```
```
        }
```
```
    }
```
```
    struct node *newNode(int data)
```
```
    {
```
```
        struct node *p = NULL;
```
```
        p = new node;
```
```
        p->data = data;
```
```
        p->left = p->right = NULL;
```
```
        return(p);
```
```
    }
```
```
};
```
```
int main()
```
```
{
```
```
    BST b1;
```
```
    struct node *root = b1.newNode(20);
```
```
    root->left = b1.newNode(8);
```
```
    root->right = b1.newNode(22);
```
```
    root->left->left = b1.newNode(4);
```
```
    root->left->right = b1.newNode(12);
```

    root->left->right->left = b1.newNode(10);

    root->left->right->right = b1.newNode(14);

```
    /* Sample tree
```
```
     *        20
```
```
     *      /    \
```
```
     *     8     22
```
```
     *    /  \
```
```
     *   4   12
```
```
     *      /  \
```
```
     *     10  14
```
```
     */
```
```
    int node_1 = 10, node_2 = 14;
```

    struct node *t = b1.lca(root, node_1, node_2);

    printf("LCA of %d and %d is %d \n", node_1, node_2, t->data);

```
    node_1 = 14, node_2 = 8;
```
```
    t = b1.lca(root, node_1, node_2);
```

    printf("LCA of %d and %d is %d \n", node_1, node_2, t->data);

```
    node_1 = 10, node_2 = 22;
```
```
    t = b1.lca(root, node_1, node_2);
```

    printf("LCA of %d and %d is %d \n", node_1, node_2, t->data);

```
    getch();
```
```
}
```

Program Explanation

1. Here in this program we have written a function to find out the lowest common ancestor of two nodes in a given tree.
2. Function lca(root,node_1,node_2) takes in three parameters which are root node of the tree, node_1 and node_2 are the two nodes whose LCA is to be determined. Function LCA returns a node therefore it is of (struct node *) type.

lca(root, node_1, node_2)
1. This function returns the lowest node who has node_1 and node_2 as its descendants or children.
2. If node_1 and node_2 lie on different sides of root i.e. (node_1 > root->data && node_2 < root->data) or vice versa, then the lca will be the root node itself.
3. In other cases such as when both node_1 and node_2 lie on left subtree i.e. (node_1 < root->data && node_2 < root->data), then the lca also lies on left subtree. So as a result we recursively call the function by passing parameters as root->left, node_1 and node_2 now.
4. By passing root->left as a parameter we go deeper and deeper inside the left subtree and return the smallest node which has both the nodes node_1 and node_2 as it’s children.
5. Similarly we do it for the right subtree by checking just one if condition and passing root->right as a parameter.

Runtime Test Cases

LCA of 10 and 14 is 12
LCA of 14 and 8 is 8
LCA of 10 and 22 is 20

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