A graph with n vertices is a tree if and only if it has n - 1 edges and has no cycles.
Data from Prob. 16
If a graph has no cycles, it must have at least 2 vertices of degree 1
Data from Prob. 18
A tree with n vertices has n - 1 edges. (Proof by induction.)

Question

A graph with n vertices is a tree if and only if it has n - 1 edges and has no cycles.

Data from Prob. 16

If a graph has no cycles, it must have at least 2 vertices of degree 1

Data from Prob. 18

A tree with n vertices has n - 1 edges. (Proof by induction.)

Accepted Answer

If G is a tree  it has no cycles  and has n - 1 edges by Prob  18  Conversely     ...

A graph with n vertices is a tree if and