  The mathematical analysis of random phylogenetic networks via analytic and
algorithmic methods has received increasing attention in the past years. In the
present work we introduce branching process methods to their study. This
approach appears to be new in this context. Our main results focus on random
level-$k$ networks with $n$ labelled leaves. Although the number of
reticulation vertices in such networks is typically linear in $n$, we prove
that their asymptotic global and local shape is tree-like in a well-defined
sense. We show that the depth process of vertices in a large network converges
towards a Brownian excursion after rescaling by $n^{-1/2}$. We also establish
Benjamini--Schramm convergence of large random level-$k$ networks towards a
novel random infinite network.
