Write an essay on trees. You should include a proof of Cayley's result on the number of labelled trees of order $n$.

Let $G$ be a graph of order $n \geq 2$. Which of the following statements are equivalent to the statement that $G$ is a tree? Give a proof or counterexample in each case.

(a) $G$ is acyclic and $e(G) \geq n-1$.

(b) $G$ is connected and $e(G) \leq n-1$.

(c) $G$ is connected, triangle-free and has at least two leaves.

(d) $G$ has the same degree sequence as $T$, for some tree $T$.