We consider the problem of the minimum number of phylogenetic trees it would take to display all splits in a given set, a problem related to k-compatibility. A set of trees that displays every single possible split is termed a universal tree set. In this note, we find the universal incompatibility U(n), the minimal size of a universal tree set for n taxa. By normalising incompatibility using U(n), one can then compare incompatibility of split systems across different numbers of taxa. We demonstrate this application by comparing two SplitsTree networks derived from archaeal genomes, with different numbers of taxa.

中文翻译：

---

关于不同分类群分裂系统不相容性的比较

我们考虑显示给定集合中所有分裂所需的最小系统发育树数量的问题，这是一个与k兼容性相关的问题。显示每一个可能的分割的一组树被称为通用树集。在本笔记中，我们找到了通用不兼容U ( n )，即n 个分类群的通用树集的最小大小。通过使用U ( n ) 对不相容性进行归一化，人们可以比较不同数量分类群的分裂系统的不相容性。我们通过比较源自古菌基因组的两个 SplitsTree 网络来演示此应用程序，它们具有不同数量的分类群。