Polynomial encoding of rooted trees with branch lengths

Phylogenetic trees are rooted trees with branch lengths that record genetic divergence or elapsed time, and quantifying differences between them is central to a wide range of evolutionary and epidemiological analyses. Graph-polynomial encodings of rooted trees provide an accurate, interpretable, and computationally efficient way to compare tree shapes, but existing polynomial encodings must be paired with auxiliary structures to study rooted trees with branch lengths. We introduce a bivariate polynomial encoding that incorporates branch lengths directly into a recursive computation from the leaf vertices to the root vertex of a tree. We prove that, for rooted trees with branch lengths and no vertices of degree two, which include all standard phylogenetic trees, two trees have the same polynomial if and only if their underlying unlabeled trees are isomorphic and the branch lengths of corresponding edges are equal. We apply the polynomial encoding to three published HIV-1 phylogenies sampled in different epidemiological settings and show that it accurately separates the three datasets based on their tree topologies and branch lengths, outperforming previous polynomial-based approaches for analyzing rooted trees with branch lengths.