标题： 关于系统发育树集合上的共凸特征 显示英文标题

标题： Coconvex characters on collections of phylogenetic trees

摘要： 在系统发育学中，一个关键问题是根据一组物种的特征集合来构建进化树，其中对于一组物种 X，特征仅仅是 X 到一组状态的函数。 在这个背景下，一个关键概念是凸性，即如果一棵树叶集为 X 的树上的特征在该树上是凸的，那么具有相同状态的叶子所跨越的子树集合是两两不相交的。 尽管在过去几十年里，单棵树上的凸特征集合已经被广泛研究，但对于共凸特征（即同时在这组树上凸的特征）知之甚少。 作为更好地理解共凸性的起点，在本文中我们证明了以下问题的一些极端结果：在所有大小为 t ≥ 2 的 n-叶树的集合上，共凸特征的最小数量是多少？如果我们将范围限制在映射到 k 种状态的共凸特征上，情况又如何？ 作为共凸性的应用，我们引入了一种新的单参数族的树度量，它介于粗粒度的罗宾逊-富兹距离和更精细的四分支距离之间。 我们表明，上述问题中的数量界限可以转化为新距离下树空间直径的界限。 我们的结果开启了几个新的有趣方向和问题，这些方向和问题有潜力应用于例如树空间和系统基因组学等领域。 显示英文摘要

摘要： In phylogenetics, a key problem is to construct evolutionary trees from collections of characters where, for a set X of species, a character is simply a function from X onto a set of states. In this context, a key concept is convexity, where a character is convex on a tree with leaf set X if the collection of subtrees spanned by the leaves of the tree that have the same state are pairwise disjoint. Although collections of convex characters on a single tree have been extensively studied over the past few decades, very little is known about coconvex characters, that is, characters that are simultaneously convex on a collection of trees. As a starting point to better understand coconvexity, in this paper we prove a number of extremal results for the following question: What is the minimal number of coconvex characters on a collection of n-leaved trees taken over all collections of size t >= 2, also if we restrict to coconvex characters which map to k states? As an application of coconvexity, we introduce a new one-parameter family of tree metrics, which range between the coarse Robinson-Foulds distance and the much finer quartet distance. We show that bounds on the quantities in the above question translate into bounds for the diameter of the tree space for the new distances. Our results open up several new interesting directions and questions which have potential applications to, for example, tree spaces and phylogenomics.