A tree is:
A. a circuitless connected graph
B. a connected graph of ‘n’ vertices with (n − 1) edges
C. a circuitless graph of ‘n’ vertices with (n − 1) edges
D. a minimally connected graph
Choose the correct answer from the options given below:

Correct option is C

A tree is a connected graph that contains no cycles (circuits). Several equivalent characterizations of a tree are given in the statements. Let us examine each one.
A. A circuitless connected graph — True
A tree is, by definition, a connected acyclic graph.
Here, "circuitless" means that the graph contains no circuit/cycle.
B. A connected graph of n vertices with (n − 1) edges — True
A fundamental property of a tree is:
​$$\boxed{|E|=|V|-1}$$​​
Thus, if a graph has n vertices and is connected with exactly n − 1 edges, it is a tree.
C. A circuitless graph of n vertices with (n − 1) edges — True
Suppose a graph has:
· n vertices
· n − 1 edges
· no circuit
A circuitless graph is a forest. If a forest has n vertices and n − 1 edges, it must have exactly one connected component.
For a forest with k connected components:
|E| = |V| - k
Here:
n – 1 = n − k
Therefore, k = 1
So, the graph is connected and acyclic, making it a tree.
D. A minimally connected graph — True
A tree is also characterized as a minimally connected graph.
This means that the graph is connected, but removing any edge from it makes the graph disconnected.
For example, A – B – C – D is a tree. If any one of its edges is removed, the graph becomes disconnected.
Information Booster
1. A tree is a connected acyclic graph.
2. If a tree has n vertices, it has exactly n – 1 edges.
3. Adding one edge to a tree creates exactly one cycle.
4. Removing any edge from a tree disconnects it.
5. A tree with n vertices has exactly n−1 edges and n−1 is the minimum number of edges required for connectivity.
Additional Knowledge
Some important equivalent conditions for a graph G to be a tree are:
​$$\boxed{\text{Connected + Acyclic}}\\\boxed{\text{Connected + }(n-1)\text{ edges}}\\\boxed{\text{Acyclic + }(n-1)\text{ edges}}\\\boxed{\text{Minimally connected}}\\$$​​
Thus, all four statements describe a tree, making Option (c) the correct answer.