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.

系统发育网络概括了系统发育树，并已被引入以描述在共存物种之间遗传物质转移的情况下的进化。有许多类的系统发育网络，都可以建模为带有标记叶子的图族。在本文中，我们专注于扎根和无根LEVEL- ķ网络并提供给定数目的叶子的有根和无根的1级和2级系统发育网络的枚举公式（精确和渐近）。我们还证明了这些网络的某些参数的分布（例如它们的循环数）是渐近正态分布的。通过首先提供我们的网络的递归描述（也称为组合规范），然后通过应用枚举，符号和解析组合的经典方法，可以获得这些结果。