Comparing phylogenetic trees of stars with spectral graph distances
Phylogenetic trees offer a hierarchical representation of chemical similarities among stars, from which their evolutionary histories can, in principle, be reconstructed. Simulation-based work suggests that phylogenetic trees built under different chemical evolution scenarios carry distinguishable structural imprints, but whether existing methods can detect this signal in practice is unclear. We evaluate distance measures, both classical tree-based and graph-theoretical, for comparing phylogenetic trees, seeking measures robust to observational noise that separate trees built from populations with distinct chemical enrichment histories. We construct Neighbor-Joining trees from chemical abundance vectors of solar twins and of stars associated with the Milky Way (MW) disk, Gaia-Sausage-Enceladus (GSE), and the Sagittarius dwarf spheroidal galaxy (Sgr). Two benchmark tests assess stability under input perturbations within the same evolutionary history and discriminative power between distinct stellar populations used as proxies for different evolutionary histories. Tree metrics rapidly saturate under perturbations of order the typical uncertainty ($\approx2σ$, or $\approx0.05$ dex per element) and fail to distinguish trees when leaf sets differ. In contrast, spectral distances vary smoothly with perturbation amplitude and provide a label-invariant method for comparing trees built from disjoint sets of stars. Among the spectral measures evaluated, the Laplacian spectral distance shows the most consistent separation across population pairs, while the normalized variant is weakest. Applied to MW, GSE, and Sgr stars, spectral measures recover a distance ordering consistent with their known chemical enrichment histories. This framework also provides a route to comparing observed enrichment histories against simulations to identify those that best reproduce the observed tree structure.