Polynomial Phylogenetic Analysis of Tree Shapes

A bstract 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 challenging aspect of 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. In this paper, we present tree comparisons and applications based on a polynomial that fully characterizes trees. Polynomials are important tools to describe discrete structures and have been used to study various objects including graphs and knots. There are also polynomials that describe rooted trees. We use tree-defining polynomials to compare tree shapes randomly generated by simulations and tree shapes reconstructed from data. Moreover, we show that the comparisons can be used to estimate parameters and to select the best-fit model that generates specific tree shapes.