Unicyclic connected graphs and vertex-deleted trees

Prove that a connected graph on \(n\) vertices has \(n\) edges if and only if it contains exactly one cycle.

Let \(n\ge 3\) and suppose that \(G\) is an \(n\)-vertex graph such that \(G\setminus\{v\}\) is a tree for every \(v\in V(G)\). Determine the number of edges in \(G\), and thereby determine \(G\).