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Tree Topologies along a Tropical Line Segment

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Abstract

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.

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Acknowledgements

R.Y. is partially supported by NSF (DMS 1916037). Also the author thank the editor and referees for improving this manuscript.

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Authors and Affiliations

  1. Naval Postgraduate School, 1411 Cunningham Road, Naval Postgraduate School, Monterey, CA, 93943-5219, USA

    Ruriko Yoshida

  2. University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Shelby Cox

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  1. Ruriko Yoshida
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  2. Shelby Cox
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Correspondence to Ruriko Yoshida.

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This paper is to dedicated to the 60th birthday of our academic groundfather, Prof. Bernd Sturmfels.

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Yoshida, R., Cox, S. Tree Topologies along a Tropical Line Segment. Vietnam J. Math. 50, 395–419 (2022). https://doi.org/10.1007/s10013-021-00526-3

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  • DOI: https://doi.org/10.1007/s10013-021-00526-3

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