Last Updated: 6 Dec, 2020

LCA Of Binary Tree

Moderate

Asked in companies

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Problem statement

You have been given a Binary Tree of distinct integers and two nodes ‘X’ and ‘Y’. You are supposed to return the LCA (Lowest Common Ancestor) of ‘X’ and ‘Y’.

The LCA of ‘X’ and ‘Y’ in the binary tree is the shared ancestor of ‘X’ and ‘Y’ that is located farthest from the root.

Note :

You may assume that given ‘X’ and ‘Y’ definitely exist in the given binary tree.

For example :

For the given binary tree

Example

LCA of ‘X’ and ‘Y’ is highlighted in yellow colour.

Input Format :

The first line contains elements of the tree in the level order form. The line consists of values of nodes separated by a single space. In case a node is null, we take -1 in its place.

The second line contains two integers ‘X’ and ‘Y’ denoting the two nodes of the binary tree.

For example, the input for the tree depicted in the below image would be :

Example

1
2 3
4 -1 5 6
-1 7 -1 -1 -1 -1
-1 -1

Explanation :

Level 1 :
The root node of the tree is 1

Level 2 :
Left child of 1 = 2
Right child of 1 = 3

Level 3 :
Left child of 2 = 4
Right child of 2 = null (-1)
Left child of 3 = 5
Right child of 3 = 6

Level 4 :
Left child of 4 = null (-1)
Right child of 4 = 7
Left child of 5 = null (-1)
Right child of 5 = null (-1)
Left child of 6 = null (-1)
Right child of 6 = null (-1)

Level 5 :
Left child of 7 = null (-1)
Right child of 7 = null (-1)

The first not-null node(of the previous level) is treated as the parent of the first two nodes of the current level. The second not-null node (of the previous level) is treated as the parent node for the next two nodes of the current level and so on.
The input ends when all nodes at the last level are null(-1).

Note :

The above format was just to provide clarity on how the input is formed for a given tree. 
The sequence will be put together in a single line separated by a single space. Hence, for the above-depicted tree, the input will be given as:

1 2 3 4 -1 5 6 -1 7 -1 -1 -1 -1 -1 -1

Output Format :

Print an integer denoting the LCA of the given binary tree nodes.

Note :

You do not need to print anything, it has already been taken care of. Just implement the given function.

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Approaches

01 Approach

The basic idea of this approach is to list down all the ancestors of the given nodes. And then, we will choose the common ancestor from the two lists which is farthest from the root of the tree.

Consider the binary tree as shown in the above figure, where we are trying to find the LCA of X and Y. Let us try to find the path from the root node to X and Y respectively and store the nodes present in the path in two separate lists pathToX and pathToY. Observe that the first few nodes(here nodes in yellow colour) in the lists will be the same which are common ancestors of node X and Y. Now, we need a common ancestor which is farthest from the root node. It is quite clear from the above image that the last common node in the lists will be the LCA.

Consider the following steps to describe the approach explained above:

Initialise two empty lists pathToX and pathToY to store the nodes in the path from the root node to ‘X’ and ‘Y’ respectively.
- Perform a pre-order traversal to find the nodes in the path from the root node to X and store it in the pathToX such that the first element in the list is the root node.
- Similarly, perform a pre-order traversal to find the nodes in the path from the root node to ‘Y’ and store it in the pathToY such that the first element in the list is the root node.
Now, start iterating from the front of the lists while the elements are equal.
The last equal element will be the LCA of X and Y.

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02 Approach

The basic idea of this approach is to find the LCA in a single traversal without using any extra space.

Let us start moving from the root node. Now consider the following situations:

If either X or Y is the root node, then the LCA of X and Y will be the root node itself because the root node is the topmost node in the binary tree.
If X and Y exist in the left subtree, then the LCA will be necessarily present in the left subtree because the farthest common ancestor from the root node will be present in the left subtree.
Similarly, if X and Y exist in the right subtree, then the LCA will be necessarily present in the right subtree.
If both X and Y are present in the different subtrees, then the LCA will be the root node.

We can easily generalize the points mentioned above for any node. Let

findLCA(TreeNode* root, int X, int Y)

be a function that returns the LCA of X and Y in the given tree or return -1 if it does not exist. Now consider the steps as follows:

If the root is NULL, then return -1 because LCA will not exist in an empty tree.
If either the root is equal to X or Y return root. In this case, the root node itself will be LCA because the root is the topmost node in the binary tree.
Let us find the LCA in the left subtree and store it in a variable leftLCA, i.e. leftLCA = findLCA(left-child of the root, X, Y)
Similarly, find the LCA in the right subtree and store it in a variable rightLCA i.e. rightLCA = findLCA(right-child of the root, X, Y)
If leftLCA and rightLCA both are not equal to -1, then the root must be the LCA. Because both ‘X’ and ‘Y’ are present in two different subtrees.
Otherwise, the LCA must be present in either of the two subtrees.
- If leftLCA is not equal to -1, return leftLCA.
- Otherwise, return rightLCA.

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