Automorphism of Tree

Question

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$.

I know every automorphism of tree is either translation or inversion or rotation. So if $\sigma$ is rotation, then we done. But i do not have any idea for cases translation or inversion.

Answer 1

This follows from $d(\theta^{-1}\sigma\theta(v),v) = d(\sigma(\theta(v)),\theta(v))$.