To determine which of these graphs are trees and which are forests, let's review the definitions of both:

1. Tree: A connected graph with no cycles. It must be a single connected component and contain no loops.
2. Forest: A collection of trees, meaning a graph that may have multiple disconnected components, each of which is a tree.

Analyzing each graph individually:

• Graph A: This graph is connected but has cycles (closed loops within the graph). Therefore, it does not meet the criteria for a tree or a forest.

  • Answer: A is neither a tree nor a forest.

• Graph B: This graph consists of multiple disconnected components, each of which is a tree (no cycles within any component). Thus, it qualifies as a forest.

  • Answer: B is a forest.

• Graph C: This is a single connected component without any cycles. Since it is connected and acyclic, it qualifies as a tree.

  • Answer: C is a tree.

• Graph D: This graph contains cycles and is also connected as a single component. Therefore, it does not qualify as either a tree or a forest.

  • Answer: D is neither a tree nor a forest.

Summary

• A: Neither tree nor forest
• B: Forest
• C: Tree
• D: Neither tree nor forest

Would you like further explanation or have any questions?

Related Questions

1. How can we determine if a graph is connected or disconnected?
2. What are the primary differences between trees and forests in graph theory?
3. Why does the presence of cycles disqualify a graph from being a tree?
4. How can spanning trees be derived from connected graphs with cycles?
5. What are some real-world applications of trees and forests in network design?

Tip

In graph theory, remember that a tree with $n$ nodes always has $n - 1$ edges, a useful property to check when identifying trees.