  We show a first-order asymptotics result for the number of galled networks
with $n$ leaves. This is the first class of phylogenetic networks of {\it
large} size for which an asymptotic counting result of such strength can be
obtained. In addition, we also find the limiting distribution of the number of
reticulation nodes of a galled networks with $n$ leaves chosen uniformly at
random. These results are obtained by performing an asymptotic analysis of a
recent approach of Gunawan, Rathin, and Zhang (2020) which was devised for the
purpose of (exactly) counting galled networks. Moreover, an old result of
Bender and Richmond (1984) plays a crucial role in our proofs, too.
