  Phylogenetic networks generalize phylogenetic trees, and have been introduced
in order to describe evolution in the case of transfer of genetic material
between coexisting species. There are many classes of phylogenetic networks,
which can all be modeled as families of graphs with labeled leaves. In this
paper, we focus on rooted and unrooted level-k networks and provide enumeration
formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2
phylogenetic networks with a given number of leaves. We also prove that the
distribution of some parameters of these networks (such as their number of
cycles) are asymptotically normally distributed. These results are obtained by
first providing a recursive description (also called combinatorial
specification) of our networks, and by next applying classical methods of
enumerative, symbolic and analytic combinatorics.
