发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推导出具有四个网状进化事件的二元系统发育网络数量的显式闭式公式，通过将79种组件图分为十组并组合已知计数实现精确枚举。
AI 中文摘要
系统发育网络为表示网状进化过程（如杂交、基因渐渗、重组和水平基因转移）提供了灵活的框架。然而，其组合复杂性使得即使是基本的枚举问题也变得困难。基于我们先前对最多具有三个网状进化事件的网络的研究，我们推导出了一个显式的闭式公式，用于计算在n个标记分类单元上具有四个网状进化事件的无限制有根二元系统发育网络的数量。我们的方法基于树组件图。我们将与具有四个网状进化事件的网络相对应的79种可能的组件图分为十组。然后，通过结合已知的单组件网络、森林以及具有较少网状进化事件的网络的计数，我们枚举了与每组相关联的网络。将这些贡献相加即可得到所需公式。该结果将无限制二元系统发育网络的精确枚举扩展到四个网状进化事件，并进一步证明了组件图在系统组织和计数日益复杂的网络类别方面的有效性。
英文摘要
Phylogenetic networks provide a flexible framework for representing reticulate evolutionary processes, such as hybridization, introgression, recombination, and horizontal gene transfer. However, their combinatorial complexity makes even basic enumeration problems difficult. Building on our previous work for networks with up to three reticulations, we derive an explicit closed-form formula for the number of unrestricted rooted binary phylogenetic networks with four reticulations on \(n\) labeled taxa. Our approach is based on tree-component graphs. We classify the 79 possible component graphs corresponding to networks with four reticulations into ten groups. We then enumerate the networks associated with each group by combining known counts of one-component networks, forests, and networks with fewer reticulations. Summing these contributions yields the desired formula. This result extends the exact enumeration of unrestricted binary phylogenetic networks to four reticulations and further demonstrates the effectiveness of component graphs for systematically organizing and counting increasingly complex network classes.
Comments18 figures. The second version added several relevant references on the enumeration of restricted classes of phylogenetic networks and updated the related work-discussion in the Introduction section. We appreciate the suggestions from Prof. Josep Batle. The main results and proofs are unchanged
