Polynomial Phylogenetic Analysis of Tree Shapes

Abstract

Phylogenetic trees are a central tool in evolutionary biology. They demonstrate evolutionary patterns among species, genes, and with modern sequencing technologies, patterns of ancestry among sets of individuals. Phylogenetic trees usually consist of tree shapes, branch lengths and partial labels. Comparing tree shapes is a main challenge in comparing phylogenetic trees as there are few tools to describe tree shapes in a quantitative, accurate, comprehensive and easy-to-interpret way. Current methods to compare tree shapes are often based on scalar indices reflecting tree imbalance, and on frequencies of small subtrees. Polynomials are important tools to describe discrete structures and have been used to study various objects including graphs and knots. There exist polynomials that describe rooted trees. In this paper, we present methods based on a polynomial that fully characterizes trees. These methods include tree metrics and machine learning tools. We use these methods to compare tree shapes randomly generated by simulations and tree shapes reconstructed from data. Moreover, we also show that the methods can be used to estimate parameters for tree shapes and select the best-fit model that generates the tree shapes.

Deep learning from phylogenies to uncover the epidemiological dynamics of outbreaks

“…Several vectorial representations of trees based either on polynomial [26,27] , Laplacian spectrum [28] or F matrices [29] have been developed previously. However, they represent the tree shape but not the branch lengths [26,27] , may lose information on trees [28] , or are specific to ultrametric trees [29] . In addition, some of these representations [27,29] require vectors or matrices of quadratic size with respect to the number of tips.…”

“…However, they represent the tree shape but not the branch lengths [26,27] , may lose information on trees [28] , or are specific to ultrametric trees [29] . In addition, some of these representations [27,29] require vectors or matrices of quadratic size with respect to the number of tips.…”

“…Several vectorial representations of trees based either on polynomial [26,27] , Laplacian spectrum [28] or F matrices [29] have been developed previously. However, they represent the tree shape but not the branch lengths [26,27] , may lose information on trees [28] .…”

“…Our CBLV tree representation could likely be improved to ease the learning phase and obtain even better parameter estimates. We tested several alternative representations, some inspired by the polynomial representation of small subtrees [26,27] , the Laplacian spectrum [28] and additive distance matrices that are equivalent to trees [50] . None was by far as convincing as CBLV, which is likely due to their large size (n 2 for distance matrices and polynomials) or numerical instabilities and potential loss of information (for Laplacian spectrum).…”

A polynomial invariant for a new class of phylogenetic networks

“…A polynomial invariant able to uniquely distinguish between rooted trees has been recently introduced in [Liu21]. Motivated to analyze and compare tree shapes in a phylogenetic context, this polynomial (which we will refer to as the Liu polynomial ) has been used both to define a similarity measure on rooted tree shapes and to estimate parameters and models via their coefficients [LBGC20]. Moreover, its generalization from trees to networks (by analyzing the set of embedded spanning trees in the network) has also been used to study the properties of randomly generated networks [JL21].…”