HMMT 2016 C9

Let $V=\{1, \ldots, 8\}$. How many permutations $\sigma: V \rightarrow V$ are automorphisms of some tree? (A graph consists of a some set of vertices and some edges between pairs of distinct vertices. It is connected if every two vertices in it are connected by some path of one or more edges. A tree $G$ on $V$ is a connected graph with vertex set $V$ and exactly $|V|-1$ edges, and an automorphism of $G$ is a permutation $\sigma: V \rightarrow V$ such that vertices $i, j \in V$ are connected by an edge if and only if $\sigma(i)$ and $\sigma(j)$ are.)