I am currently working with the automorphisms of a rooted tree $T$, and I have a question regarding $\text{Aut}(T)$ as a profinite group. Recall that we can make $\text{Aut}(T)$ into a profinite group by choosing the subgroups $\text{St}_{\text{Aut}(T)}(n)$ as a base of open neighbourhoods of the identity.
I have to consider a closed subgroup $G \leq \text{Aut}(T)$. I read that, since $G$ is closed, then it is uniquely determined by its actions on all finite levels of the tree $T$.
Why does this hold?
Thank you for your help.