MacArthur, Sanchez-Garcia, and Anderson have used the ratio of the order of $|Aut(G)|$ and $n!$ (i.e., order of $S_n$) as a normalized measure of the symmetries present in a graph.
I am working on a project where the symmetries of rooted trees are important, and am trying to calculate the analogue of $|S_n|$ -- i.e., the order of the largest possible automorphism group on a rooted tree with $n$ vertices. I am a relative beginner to algebraic graph theory and after some time spent attempting to find the answer in the literature, I am hoping someone can provide a timely pointer.
Thank you!