Question

Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree.According to the definition of LCA on Wikipedia: "The lowest common ancestor is defined between two nodes p and q as the lowest node in T that has both p and q as descendants (where we allow a node to be a descendant of itself).

_______3______
/ \
___5__ ___1__
/ \ / \
6 _2 0 8
/ \
7 4

Answer 1

To find the lowest common ancestor in a binary tree, we start from the root and recursively traverse its subtrees, checking for the nodes in question. If both p and q are found in different subtrees, the current node is their LCA; otherwise, the search continues in the subtree that contains either p or q.

Step-by-step explanation:

The question revolves around finding the lowest common ancestor (LCA) in a binary tree. Calculating the LCA of two nodes (p and q) involves traversing the tree from the root while maintaining a few simple rules:

1. Start from the root of the tree.
2. If the current node is either p or q, return it as the LCA.
3. Recurse into the left and the right subtrees looking for nodes p and q.
4. If both the left and the right recursions return non-NULL values, it means that the current node is the LCA.
5. If only one child returns a non-NULL value, that child might contain both p and q and is thus the LCA.

This approach relies on the recursive nature of the tree structure and allows us to effectively find the LCA with a time complexity of O(n), where n is the number of nodes in the tree.

Answer 2

The lowest common ancestor (LCA) of nodes 5 and 1 in the given binary tree is node 3.

Step-by-step explanation:

In the given binary tree, the LCA problem aims to find the node that is the lowest common ancestor of two given nodes. In this case, the nodes provided are 5 and 1. To solve this problem, we'll traverse the tree using a recursive approach.

Starting from the root node 3, we check if either of the given nodes (5 or 1) matches the current node. Since neither 5 nor 1 is equal to 3, we move to its left child (5) and right child (1). At this point, both 5 and 1 are found on different subtrees.

As the LCA should be the lowest node having both nodes 5 and 1 as descendants, we continue our search downward. Traversing further, we find that node 3 satisfies this condition. Node 3 is the lowest common ancestor for nodes 5 and 1 as both nodes can be found in its subtrees.

Hence, after the traversal, we determine that the lowest common ancestor (LCA) of nodes 5 and 1 in the given binary tree is indeed node 3. This solution is achieved by traversing through the tree and identifying the node that satisfies the conditions of being the lowest common ancestor for the given nodes.