Lowest Common Ancestor (LCA) Concepts in Trees Quiz Quiz

Explore fundamental concepts of the Lowest Common Ancestor (LCA) in tree data structures with straightforward, scenario-based questions designed to solidify your understanding. This quiz covers LCA definitions, algorithms, binary trees, and common pitfalls—ideal for learners and anyone reviewing tree algorithms.

1. Identifying LCA in a Simple Binary Tree

   In a binary tree, if node X is an ancestor of nodes Y and Z, which node is the lowest common ancestor (LCA) of Y and Z?

   1. Y
   2. X
   3. Z
   4. Root

   Explanation: X is the lowest node in the tree that is an ancestor of both Y and Z, satisfying the definition of an LCA. Y and Z are descendants, not ancestors, so they cannot be the LCA. Root refers to the tree's topmost node, which is only the LCA if there is no lower common ancestor. In this scenario, X is the correct answer.

2. LCA Definition Understanding

   Which statement best defines the lowest common ancestor (LCA) in a rooted tree?

   1. It is any common ancestor of the given nodes.
   2. It is the closest leaf node to the given nodes.
   3. It is the deepest node that is an ancestor of both given nodes.
   4. It is the node with the highest value among the given nodes.

   Explanation: The LCA is by definition the deepest (i.e., farthest from the root) node that is an ancestor to both nodes in question. Any common ancestor would include less specific, higher ancestors. The closest leaf is unrelated, and the node with the highest value is not relevant to the LCA concept. Therefore, the first option expresses the correct definition.

3. LCA in a Chain

   Suppose you have a tree where each node has exactly one child (a straight line). What is the LCA of the bottom-most node and the root?

   1. The second node
   2. The bottom-most node
   3. There is no LCA
   4. The root

   Explanation: In a straight line or chain tree, the root is an ancestor of all nodes, and thus the LCA of any two nodes is the higher (closer to root) one. The second node and bottom-most node are not common ancestors of both nodes. There is always an LCA in trees, so 'there is no LCA' is incorrect. Thus, the correct answer is the root.

4. LCA of a Node with Itself

   If you are asked to find the LCA of a node and itself in a binary tree, what is the result?

   1. There is no LCA in this case
   2. The leftmost node
   3. The root node
   4. The node itself

   Explanation: By definition, the LCA of a node with itself is the node itself, since it is its own ancestor at depth zero. The root node is only returned if the target node is the root. The leftmost node has no special significance here, and an LCA always exists for a node with itself, so 'there is no LCA in this case' is incorrect.

5. Binary Search Tree (BST) LCA Scenario

   In a binary search tree, the values 8 and 12 are in the left and right subtrees of node 10, respectively. What is their lowest common ancestor?

   1. 10
   2. 8
   3. 12
   4. Root of the tree

   Explanation: Node 10 is the first node where the paths to 8 and 12 diverge, making it their LCA. Nodes 8 and 12 are not ancestors to each other, so they can't be the LCA. The root would only be correct if there were no lower common ancestor, but in this case, 10 is lower. Therefore, 10 is correct.

6. Multiple Paths in Non-Binary Tree

   In a tree with multiple children per node, what does the LCA of two nodes depend on?

   1. The values stored in the nodes
   2. Whether the tree is balanced
   3. The number of children per node
   4. Their depth and ancestral paths in the tree

   Explanation: The LCA is determined by where the paths from each node to the root intersect deepest, which relates to their depth and ancestor paths. The number of children or balance of the tree does not directly affect the LCA definition. Node values are only relevant in BST algorithms but do not define the LCA itself.

7. Unique LCA Property

   Is it possible for two different nodes in a tree to have more than one lowest common ancestor?

   1. Yes, if the tree contains cycles.
   2. No, unless the tree is empty.
   3. Yes, if the tree is not binary.
   4. No, there is always exactly one LCA for any two nodes in a tree.

   Explanation: A tree's structure ensures that any two nodes share exactly one unique lowest common ancestor. Cycles are not allowed in trees, so having multiple LCAs is not possible. Whether the tree is binary or not does not affect this property, nor does an empty tree scenario since there are no nodes.

8. Ancestor Relationships

   If node Q is an ancestor of node R in a tree, what is the LCA of Q and R?

   1. R
   2. Root
   3. Q
   4. Their parent node

   Explanation: Whenever one node is an ancestor of another, the ancestor itself (Q) is considered the LCA. R is not an ancestor of Q, so it cannot be the LCA. The parent node of either is not always applicable if Q is the direct ancestor. The root is only the LCA if Q is the root.

9. Algorithm Choice

   Which method is commonly used to find the LCA in trees without parent pointers?

   1. Bubble sort
   2. Depth-first search traversal
   3. Breadth-first search only
   4. Insertion sort

   Explanation: Depth-first search allows you to record ancestor paths from the root, helping to compare the paths for two nodes to determine the LCA. Bubble sort and insertion sort are unrelated sorting algorithms. While breadth-first search can be used, it is not typically chosen over depth-first traversal for this purpose in trees.

10. Input Order in LCA Function

    If you call an LCA function with two node inputs, does the order of nodes affect the result?

    1. Yes, the result will differ if inputs are swapped.
    2. No, provided the nodes are leaf nodes.
    3. Yes, the first node is always considered the LCA.
    4. No, the order does not affect the LCA.

    Explanation: The LCA of two nodes is defined by their position in the tree, not the input order; it will always be the same regardless of how the nodes are supplied. The first node is not automatically taken as the LCA. The result will not differ when swapping the order. Leaf status is irrelevant to this property.