Palm tree space is the set of phylogenetic trees on a fixed leaf set, represented as pairwise leaf-distance vectors in the tropical projective torus and equipped with the tropical (generalized Hilbert projective) metric. Because this metric space is proven complete and separable — a Polish space — probability measures, Frechet means and variances, and parametric statistical inference on collections of trees are all well defined on it, which the non-Euclidean BHV geodesic tree space does not fully support because its Frechet means stick to lower-dimensional strata. Students learn how a metric choice determines which statistical and machine-learning methods are available on a non-Euclidean data space, and how the tropical line segment gives geodesics computable in O(N^2 log N) versus O(N^4) for BHV geodesics.
This arXiv paper by Anthea Monod, Bo Lin, Ruriko Yoshida, and Qiwen Kang directly addresses the mathematical foundations for phylogenetic taxonomy and cladistics using tropical geometry. The paper de…