  We study the size of the automorphism group of two different types of random
trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove
that it asymptotically follows a log-normal distribution and provide asymptotic
formulas for mean and variance of the logarithm of the size of the automorphism
group. While the proof for Galton--Watson trees mainly relies on probabilistic
arguments and a general result on additive tree functionals, generating
functions are used in the case of rooted P\'olya trees. We also show how to
extend the results to some classes of unrooted trees.
