What are the automorphisms of an $n$-regular tree?

Question

Let $T$ be the connected tree in which each vertex has $n$ neighbors. (So $T$ is infinite.) What is the full automorphism group of $T$?

Answer 1

$T$ itself is the Cayley graph of the group $A$ with presentation $$\langle g_1,\dots,g_n|g_i^2\rangle$$ An automorphism of $T$ is specified by the image of the identity vertex or "root", then the permutations of all subtrees starting from the root image. Attached to the root image are $n$ subtrees $B$ rooted at a vertex of degree $n-1$, which in turn have $n-1$ maximal subtrees isomorphic to $B$. Thus in a manner like the Rubik's Cube group $$\newcommand{A}{\operatorname{Aut}}\A(T)\cong(\A(B)\wr S_n)\rtimes A$$ $$\cong((\A(B)\wr S_{n-1})\wr S_n)\rtimes A$$ $$\cong(((\dots\wr S_{n-1})\wr S_{n-1})\wr S_n)\rtimes A$$