A note on the metric properties of trees

Graph theory

Harary, F. (1969) · Cited by 10,739

Mathematics in the Archaeological and Historical Sciences

(1970) · Bulletin of the London Mathematical Society · doi:10.1112/blms/2.2.252 · Cited by 388

Mathematics in the Archaeological and Historical Sciences.

Sneath, P. H. A., Hodson, F. R., Kendall, D. G., & Tautu, P. (1972) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2344692 · Cited by 313

The structure and construction of taxonomic hierarchies

JARDINE, C. J., Jardine, N., & Sibson, R. (1967) · Mathematical Biosciences · doi:10.1016/0025-5564(67)90032-6 · Cited by 123

A note on the tree realizability of a distance matrix

Pereira, J. M. S. S. (1969) · Journal of Combinatorial Theory · doi:10.1016/s0021-9800(69)80092-x · Cited by 103

A convergence result on random products of mappings in metric trees

Al-Mezel, S. A., & Khamsi, M. A. (2012) · Fixed Point Theory and Applications · doi:10.1186/1687-1812-2012-57

References

(1989) · Elsevier eBooks · doi:10.1016/b978-0-12-425402-2.50014-4

Special Metrics

Simovici, D. A., & Djeraba, C. (2014) · Advanced information and knowledge processing · doi:10.1007/978-1-4471-6407-4_14

Shortest Paths

Jungnickel, D. (2012) · Algorithms and computation in mathematics · doi:10.1007/978-3-642-32278-5_3

References

(2007) · Elsevier eBooks · doi:10.1016/b978-012088794-1/50015-x

Metric, Ultra

Rohlf, F. J. (2013) · Elsevier eBooks · doi:10.1016/b978-0-12-374984-0.00936-0

A note on generalized four-point inequality

Petrov, E., & Salimov, R. R. (2023) · Ukrainian Mathematical Bulletin · doi:10.37069/1810-3200-2023-20-1-6

Clustering Methods Over the Tropically Convex SetsPreprint

Barnhill, D., & Yoshida, R. (2023) · Research Square · doi:10.21203/rs.3.rs-3047827/v1

Phylogenetic Analysis of Reticulate Software Evolution

Mori, A., & Hashimoto, M. (2023) · International Conference on Mining Software Repositories (MSR) · doi:10.1109/msr59073.2023.00074

Relation between broadcast domination and multipacking numbers on chordal and other hyperbolic graphs

Das, S., Foucaud, F., Islam, S. S., & Mukherjee, J. (2026) · Discrete Applied Mathematics · doi:10.1016/j.dam.2026.02.006

The Four Point Condition: An Elementary Tropicalization of Ptolemy’s Inequality

Gómez, M., & Mémoli, F. (2023) · American Mathematical Monthly · doi:10.1080/00029890.2023.2285695 · Cited by 3

Existence of non-unique neighbor-joining phylogenetic trees of microsatellite markersPreprint

Segura-Alabart, N., Serratosa, F., & Fernández, A. (2023) · Research Square · doi:10.21203/rs.3.rs-3786817/v1

Additional file 2 of From pairs of most similar sequences to phylogenetic best matches

Stadler, P. F., Geiß, M., Schaller, D., Sánchez, A. L., Laffitte, M. E. G., Valdivia, D. I., Hellmuth, M., & Hernández-Rosales, M. (2020) · Figshare · doi:10.6084/m9.figshare.12109566

Four-point condition matrices of edge-weighted trees

Azimi, A., Jana, R., Nagar, M. K., & Sivasubramanian, S. (2024) · Special Matrices · doi:10.1515/spma-2024-0011

Admixed populations in the neighbor-joining algorithm: a geometric analysis with five taxaPreprint

Zhang, J. Z., Lo, W. T. ‘., Stillman, M., & Kim, J. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.10.18.619141

Bayesian Least-Squares Supertrees (BLeSS): Flexible Inference of Large Time-Calibrated PhylogeniesPreprint

Černý, D., & Slater, G. J. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.11.29.625936

Unimodality and peak location of the characteristic polynomials of two distance matrices of trees

Jana, R., Mahato, I., & Sivasubramanian, S. (2025) · Computational and Applied Mathematics · doi:10.1007/s40314-025-03163-5

Projected Gradient Descent Method for Tropical Principal Component Analysis over Tree Space

Yoshida, R. (2025) · Mathematics · doi:10.3390/math13111776

Invariants recovering the reduction type of a hyperelliptic curve

Kellock, L. C., & García, E. L. (2025) · Journal of Symbolic Computation · doi:10.1016/j.jsc.2025.102533

Tropical cluster varieties of type C

Makhlin, I. Y. (2026) · International Mathematics Research Notices · doi:10.1093/imrn/rnag193

Additional file 1 of From pairs of most similar sequences to phylogenetic best matches

Stadler, P. F., Geiß, M., Schaller, D., Sánchez, A. L., González-Laffitte, M., Valdivia, D. I., Hellmuth, M., & Hernández-Rosales, M. (2020) · Figshare · doi:10.6084/m9.figshare.12109560

Probability metrics for tropical spaces of different dimensions

Talbut, R., Tramontano, D., Cao, Y., Drton, M., & Monod, A. (2025) · Foundations of Data Science · doi:10.3934/fods.2026001 · Cited by 2

DecoDiPhy: Deconvolving Phylogenetic Distance MixturesPreprint

Arasti, S., Şapcı, A. O. B., Rachtman, E., El-Kebir, M., & Mirarab, S. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.01.18.700179

Tropical Fermat–Weber Points over Spaces of M-Ultrametrics

Cox, S., Sabol, J., Talbut, R., & Yoshida, R. (2026) · Vietnam Journal of Mathematics · doi:10.1007/s10013-026-00799-6

Bounds on the Sequence Length Sufficient to Reconstruct Binary Level-1 Phylogenetic Networks Under the CFN Model

Frohn, M., Holtgrefe, N., van Iersel, L. J. J., Jones, M., & Kelk, S. M. (2026) · Annals of Combinatorics · doi:10.1007/s00026-026-00830-0

Minimum Deviation Distance Realization

Bar-Noy, A., Peleg, D., Perry, M., Ran, Y., & Rawitz, D. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-26465-7_5

Clean Breaks: Attention Force Field Deconvolution of Recombination Across Sequence ManifoldsPreprint

Pond, S. L. K., Callan, D., Weaver, S., Verdonk, H., Zehr, J., Nekrutenko, A., Paisie, T. K., & Martin, D. P. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.10.19.760237

The Minimum Evolution Criterion of Phylogeny Estimation via Linear Programming: Theoretical Foundations and Improved Solution Approaches

Catanzaro, D., Aringhieri, R., Summa, M. D., & Pesenti, R. (2015) · Dépôt institutionnel de l'Université libre de Bruxelles (Université Libre de Bruxelles)

c © Birkhauser Verlag, Basel, 2006 Annals of Combinatorics Characterization of the Distance between Subtrees of a Tree by the Associated Tight Span

Hirai, H. (2004)

Characterization of the Distance between Subtrees of a Tree by the Associated Tight Span

(2004)

On the sum of the squares of all distances in trees with fixed diameter.

Geng, X., & Zhao, H. (2018) · Ars Combinatoria

Extremal values for ratios of distances in trees

Barefoot, C. A., Entringer, R. C., & Székely, Ł. A. (1997) · Discrete Applied Mathematics · doi:10.1016/s0166-218x(97)00068-1 · Cited by 41

Distance defined by spanning trees in graphs

Chartrand, G., Nebeský, L., & Zhang, P. (2007) · Discussiones Mathematicae Graph Theory · doi:10.7151/dmgt.1375 · Cited by 2