Dickey Emily H, Rosenberg Noah A
Department of Biology, Stanford University, Stanford, CA 94305, USA.
Philos Trans R Soc Lond B Biol Sci. 2025 Feb 13;380(1919):20230307. doi: 10.1098/rstb.2023.0307. Epub 2025 Feb 20.
In mathematical models of phylogenetic trees evolving in time, a labelled history for a rooted labelled bifurcating tree is a temporal sequence of the branchings that give rise to the tree. That is, given a leaf-labelled tree with [Formula: see text] leaves and [Formula: see text] internal nodes, a labelled history is an identification between the internal nodes and the set [Formula: see text], such that the label assigned to a given node is strictly greater than the labels assigned to its descendants. We generalize the concept of labelled histories to [Formula: see text]-furcating trees. Consider a rooted labelled tree in which each internal node has exactly [Formula: see text] children, [Formula: see text]. We first generalize the enumeration of labelled histories for a bifurcating tree ([Formula: see text]) to enumerate labelled histories for an [Formula: see text]-furcating tree with arbitrary [Formula: see text]. We formulate a conjecture for the rooted unlabelled [Formula: see text]-furcating tree shape on [Formula: see text] internal nodes whose labelled topologies have the most labelled histories. Finally, we enumerate labelled histories for [Formula: see text]-furcating trees in a setting that allows for simultaneous branchings. These results advance mathematical phylogenetic modelling by extending computations concerning fundamental features of bifurcating phylogenetic trees to a more general class of multifurcating trees.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.
在随时间演化的系统发育树的数学模型中，有根带标签二叉树的带标签历史是产生该树的分支的时间序列。也就是说，给定一棵有(n)个叶子节点和(n - 1)个内部节点的带叶子标签的树，一个带标签历史是内部节点与集合({1, 2, \ldots, n - 1})之间的一种对应关系，使得分配给给定节点的标签严格大于分配给其后代的标签。我们将带标签历史的概念推广到(k)叉树。考虑一棵有根带标签的树，其中每个内部节点恰好有(k)个孩子，(k \geq 2)。我们首先将二叉树（(k = 2)）的带标签历史的枚举进行推广，以枚举具有任意(k)的(k)叉树的带标签历史。我们针对具有最多带标签历史的带标签拓扑结构的、有(n)个内部节点的无根无标签(k)叉树形状提出一个猜想。最后，我们在允许同时分支的情况下枚举(k)叉树的带标签历史。这些结果通过将关于二叉系统发育树基本特征的计算扩展到更一般的多叉树类别，推进了数学系统发育建模。本文是主题为“进化的数学理论”：可追溯到100年前的系统发育模型的一部分。
