If $T$ is a tree, and $T$ has an odd number of vertices, then $\forall f$, where $f$ is automorphism $\Rightarrow \exists$ fixed point (vertex). What it means:
Formally, an automorphism of a tree $T$ is a permutation $\sigma$ of the vertex set $V$ of tree, such that the pair of vertices $(u, v)$ form an edge if and only if the pair $(\sigma(u),\sigma(v))$ also form an edge.
If $T$ is a tree with vertices ($v1,..vn$), then after applying $f$, some vertex will be always stay in place.
if somebody want to illustrate this, I will do.