Given a Binary Search Tree (BST), How do we find the Lowest Common Ancestor (LCA) of the given two nodes?

For example in the following BST, the LCA of 5 and 8 is 7 because it is the nearest ancestor common to both the nodes.

Let us assume that the LCA(5,7) is 7 itself ( i.e We can consider that a node is a ancestor of itself).

4

/ \

2 7

/ \

5 9

/

8

We start at the root node and check if the data at root node lies between the given two nodes. If it is we have found the LCA. If the root data is lesser than the given two nodes, we ignore the left sub-tree and continue with the right-sub tree. If the root data is greater than given two nodes, we ignore the right sub-tree and repeat the same procedure with the left sub-tree.

This approach assumes that the given two nodes are present in the tree.

It runs in O( n log n) on average. and O(n) in worst case( skewed BST).

Following is the Java code which implements the above algorithm.

For example in the following BST, the LCA of 5 and 8 is 7 because it is the nearest ancestor common to both the nodes.

Let us assume that the LCA(5,7) is 7 itself ( i.e We can consider that a node is a ancestor of itself).

4

/ \

2 7

/ \

5 9

/

8

We start at the root node and check if the data at root node lies between the given two nodes. If it is we have found the LCA. If the root data is lesser than the given two nodes, we ignore the left sub-tree and continue with the right-sub tree. If the root data is greater than given two nodes, we ignore the right sub-tree and repeat the same procedure with the left sub-tree.

This approach assumes that the given two nodes are present in the tree.

It runs in O( n log n) on average. and O(n) in worst case( skewed BST).

Following is the Java code which implements the above algorithm.

```
/**
* Created with IntelliJ IDEA.
* User: Ravi
* Date: 9/27/13
* Time: 10:58 PM
* To change this template use File | Settings | File Templates.
*/
public class Main {
public static void main(String[] args)
{
BinarySearchTree tree = new BinarySearchTree();
tree.insert(4);
tree.insert(8);
tree.insert(2);
tree.insert(3);
tree.insert(1);
tree.insert(5);
tree.insert(9);
System.out.println("The lease common ancestor of 9 and 1 is " + tree.findLCA(9,1).getData()) ;
System.out.println("The lease common ancestor of 4 and 8 is " + tree.findLCA(4,8).getData()) ;
}
//Define Binary Search Tree Node
static class BSTNode
{
public BSTNode(int d )
{
data = d;
left = null;
right = null;
}
public BSTNode(int d, BSTNode l, BSTNode r)
{
data = d;
left = l;
right = r;
}
public int getData()
{
return data;
}
public void setData(int d)
{
data = d;
}
public BSTNode getLeft()
{
return left;
}
public void setLeft(BSTNode l)
{
left = l;
}
public BSTNode getRight()
{
return right;
}
public void setRight(BSTNode r)
{
right = r;
}
private int data; //data
private BSTNode left; //left subtree
private BSTNode right; //right subtree
}
//Define Binary Search Tree
static class BinarySearchTree
{
public BinarySearchTree()
{
root = null;
}
public BSTNode getRoot()
{
return root;
}
//Finds the least common ancestor for the given two nodes
public BSTNode findLCA(int a, int b)
{
//Find the two nodes with given values
BSTNode nodeA = find(a);
BSTNode nodeB = find(b);
BSTNode r = root;
while ( r != null )
{
if( r.getData() < a && r.getData() < b )
r = r.getRight();
else if( r.getData() > a && r.getData() > b)
r = r.getLeft();
else
break;
}
return r;
}
//Finds the BST node with the given data
public BSTNode find(int d)
{
BSTNode r = root;
while ( r != null && r.getData() != d )
{
if( r.getData() == d )
break;
else if( r.getData() < d )
r = r.getRight();
else
r = r.getLeft();
}
return r;
}
//inserts the given data into a BST
public void insert(int d)
{
if( root == null )
{
root = new BSTNode(d);
}
else
{
BSTNode current = root;
BSTNode prev = null;
while( current != null )
{
prev = current;
if( current.getData() < d )
{
current = current.getRight();
}
else
{
current = current.getLeft();
}
}
if( d < prev.getData() )
{
prev.setLeft( new BSTNode(d));
}
else
{
prev.setRight(new BSTNode(d));
}
}
}
private BSTNode root;
}
}
```