Analyzing Phylogenetic Trees with a Tree Lattice Coordinate System and a Graph Polynomial

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 unlabeled 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 article, we introduce representation and comparison methods for rooted unlabeled 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. [Graph polynomial; likelihood-free inference; phylogenetics; tree lattice; tree metrics.].