Phylogenetic trees are a central tool in many areas of life science and medicine. They demonstrate evolutionary patterns among species, genes, and patterns of ancestry among sets of individuals. The tree shapes and branch lengths of phylogenetic trees encode evolutionary and epidemiological information. To extract information from tree shapes and branch lengths, representation and comparison methods for phylogenetic trees are needed. Representing and comparing tree shapes and branch lengths of phylogenetic trees are challenging, for a tree shape is unlabelled and can be displayed in numerous different forms, and branch lengths of a tree shape are specific to edges whose positions vary with respect to the displayed forms of the tree shape. In this paper, we introduce representation and comparison methods for rooted unlabelled phylogenetic trees based on a tree lattice that serves as a coordinate system for rooted binary trees with branch lengths and a graph polynomial that fully characterizes tree shapes. We show that the introduced tree representations and metrics provide distance-based likelihood-free methods for tree clustering, parameter estimation and model selection, and apply the methods to analyze phylogenies reconstructed from virus sequences.

中文翻译：

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用树格坐标系和图多项式分析系统发育树

系统发育树是生命科学和医学许多领域的核心工具。它们展示了物种、基因之间的进化模式，以及个体之间的祖先模式。系统发育树的树形和分支长度编码进化和流行病学信息。为了从树的形状和分支长度中提取信息，需要系统发育树的表示和比较方法。表示和比较系统发育树的树形和分支长度具有挑战性，因为树形是未标记的，并且可以以多种不同的形式显示，并且树形的分支长度特定于边缘，其位置相对于显示的形式不同树的形状。在本文中，我们介绍了基于树格的有根未标记系统发育树的表示和比较方法，该树格用作具有分支长度的有根二叉树的坐标系和完全表征树形状的图多项式。我们表明，引入的树表示和度量为树聚类、参数估计和模型选择提供了基于距离的无似然方法，并将这些方法应用于分析从病毒序列重建的系统发育。