Hostname: page-component-54dcc4c588-m259h Total loading time: 0 Render date: 2025-09-16T04:27:28.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Can you tell a tree from its branches?

Part of: Graph theory

Published online by Cambridge University Press:  15 September 2025

Chengyang Qian Open the ORCID record for Chengyang Qian [Opens in a new window] ,
Yaokun Wu Open the ORCID record for Yaokun Wu [Opens in a new window]  and
Min Xie Open the ORCID record for Min Xie [Opens in a new window]
Chengyang Qian
Affiliation:
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
Yaokun Wu*
Affiliation:
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
Min Xie
Affiliation:
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
*
Corresponding author: Yaokun Wu, email: ykwu@sjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

For each set X, an X-split is a partition of X into two parts. For each X-split S and each subset $Y\subseteq X$, the restriction of S on Y is the Y-split whose parts are obtained by intersecting the parts of S with Y. For a graph G with vertex set V, the G-coboundary size of a V-split S is the number of edges in G having non-empty intersections with both parts of S. Let T be a tree without degree-two vertices, and let V and L denote its vertex set and leaf set, respectively. For each positive integer k, a k-split on T is an L-split that is the restriction of a V-split with T-coboundary size k, while a score-k split on T is a k-split on T that is not any k′-split for any integer $k' \lt k$. Buneman’s split equivalence theorem states that the tree T is entirely encoded by its system of score-1 splits. We identify the unique exceptional case in which the tree T is not determined by its score-2 split system. To explore how our work can be extended to more general tree isomorphism problems, we propose several conjectures and open problems related to set systems and generalized Buneman graphs.

Keywords

2-special configuration(k, 2)-hypergraphsmallest 2-extension

MSC classification

Primary: 05C05: Trees
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05C85: Graph algorithms 92D15: Problems related to evolution

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 35
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adam-Day, S., On the continuous gradability of the cut-point orders of $\mathbb{R}$-trees, Topology Appl. 306 (2022), 107937, pp. 21. doi:10.1016/j.topol.2021.107937.Google Scholar
Aivaliotis, M., Gordon, G. and Graveman, W., When bad things happen to good trees, J. Graph Theory 37(2): (2001), 79–99. doi:10.1002/jgt.1004.CrossRefGoogle Scholar
Aliste-Prieto, J., Martin, J. L., Wagner, J. D. and Zamora, J., Chromatic symmetric functions and polynomial invariants of trees, Bull. Lond. Math. Soc. 56(11): (2024), 3452–3476. doi:10.1112/blms.13144.CrossRefGoogle Scholar
Backelin, J. and Linusson, S., Parity splits by triple point distances in X-trees, Ann. Comb. 10(1): (2006), 1–18. doi:10.1007/s00026-006-0270-1.Google Scholar
Baez, J. C. and Otter, N., Operads and phylogenetic trees, Theory Appl. Categ. 32 (2017), 1397–1453, www.tac.mta.ca/tac/volumes/32/40/32-40abs.html.Google Scholar
Bandelt, H.-J. and Röhl, A., Quasi-median hulls in Hamming space are Steiner hulls, Discrete Appl. Math. 157(2): (2009), 227–233. doi:10.1016/j.dam.2006.09.015.Google Scholar
Bertrand, P. and Diatta, J., An interval convexity-based framework for multilevel clustering with applications to single-linkage clustering, Discrete Appl. Math. 342 (2024), 38–63. doi:10.1016/j.dam.2023.08.003.Google Scholar
Besenbacher, S., Mailund, T., Westh-Nielsen, L. and Pedersen, C. N. S., RBT – A tool for building refined Buneman trees, Bioinformatics 21(8): (2004), 1711–1712. doi:10.1093/bioinformatics/bti195.Google ScholarPubMed
Bordewich, M., Deutschmann, I. M., Fischer, M., Kasbohm, E., Semple, C. and Steel, M., On the information content of discrete phylogenetic characters, J. Math. Biol. 77(3): (2018), 527–544. doi:10.1007/s00285-017-1198-2.CrossRefGoogle ScholarPubMed
Bordewich, M., Semple, C. and Steel, M., Identifying X-trees with few characters, Electron. J. Combin. 13(1): (2006), R83, pp. 11. doi:10.37236/1109.Google Scholar
Bowler, N. and Kneip, J., Separations of sets, Order 37(2): (2020), 411–425. doi:10.1007/s11083-019-09512-4.CrossRefGoogle Scholar
Buneman, P., The recovery of trees from measures of dissimilarity. Mathematics in the Archeological and Historical Sciences, In Hodson, F., Kendall, D. Tautu, P., (Eds.) pp. 387–395 (Edinburgh University Press, United Kingdom, 1971). https://homepages.inf.ed.ac.uk/opb/homepagefiles/phylogeny-scans/manuscripts.pdf.Google Scholar
Burghart, F., Polynomial invariants for rooted trees related to their random destruction, Electron. J. Combin. 31(4): (2024), P4.37, pp. 19. doi:10.37236/11975.Google Scholar
Cantone, D., Doignon, J. -P., Giarlotta, A. and Watson, S., Resolutions of convex geometries, Electron. J. Combin. 28(4): (2021), P4.26, pp. 39. doi:10.37236/10328.Google Scholar
Casanellas, M. and Fernández-Sánchez, J., Relevant phylogenetic invariants of evolutionary models, J. Math. Pures Appl. (9) 96(3): (2011), 207–229. doi:10.1016/j.matpur.2010.11.002.Google Scholar
Chalopin, J., Chepoi, V. and Giocanti, U., Graphs with convex balls, Geom. Dedicata 217(4): (2023), 67, pp. 52. doi:10.1007/s10711-023-00803-0.CrossRefGoogle Scholar
Chaudhary, S. and Gordon, G., Tutte polynomials for trees, J. Graph Theory 15(3): (1991), 317–331. doi:10.1002/jgt.3190150308.CrossRefGoogle Scholar
Choe, Y.-B., Huber, K. T., Koolen, J. H., Kwon, Y. S. and Moulton, V., Counting vertices and cubes in median graphs of circular split systems, European J. Combin. 29(2): (2008), 443–456. doi:10.1016/j.ejc.2007.02.003.Google Scholar
Danilov, V. I., Karzanov, A. V. and Koshevoy, G. A., On interrelations between strongly, weakly and chord separated set-systems (a geometric approach), J. Algebraic Combin. 54(4): (2021), 1299–1327. doi:10.1007/s10801-021-01047-5.CrossRefGoogle Scholar
Dasarathy, G., Mossel, E., Nowak, R. and Roch, S., A stochastic Farris transform for genetic data under the multispecies coalescent with applications to data requirements, J. Math. Biol. 84(5): (2022), 36, pp. 37. doi:10.1007/s00285-022-01731-5.CrossRefGoogle ScholarPubMed
Develin, M. and Sturmfels, B., Tropical convexity, Doc. Math. 9 (2004), 1–27. doi:10.4171/DM/154.Google Scholar
Diestel, R., Hundertmark, F. and Lemanczyk, S., Profiles of separations: in graphs, matroids, and beyond, Combinatorica 39(1): (2019), 37–75. doi:10.1007/s00493-017-3595-y.CrossRefGoogle Scholar
Dovgoshey, O. and Kostikov, A., Locally finite ultrametric spaces and labeled trees, J. Math. Sci. New York 276(5): (2023), 614–637. doi:10.1007/s10958-023-06786-3.CrossRefGoogle Scholar
Dress, A. W. M., K. T., Huber, and Moulton, V.. Some variations on a theme by Buneman, Ann. Comb. 1(4): (1997), 339–352. doi:10.1007/BF02558485.CrossRefGoogle Scholar
Dress, A. W. M., K. T., Huber, and Moulton, V.. A comparison between two distinct continuous models in projective cluster theory: The median and the tight-span construction, Ann. Comb. 2(4): (1998), 299–311. doi:10.1007/BF01608527.Google Scholar
Dress, A. W. M., Hendy, M. D., Huber, K. T. and Moulton, V., On the number of vertices and edges of the Buneman graph, Ann. Comb. 1(4): (1997), 329–337. doi:10.1007/BF02558484.CrossRefGoogle Scholar
Dress, A. W. M., Huber, K. T., Koolen, J., Moulton, V. and Spillner, A., Basic Phylogenetic Combinatorics., (Cambridge University Press, Cambridge, 2012) doi:10.1017/CBO9781139019767.Google Scholar
Dress, A. W. M., Huber, K. T. and Moulton, V., An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex, Adv. Math. 168(1): (2002), 1–28. doi:10.1006/aima.2001.2039.CrossRefGoogle Scholar
Dress, A. W. M., Huber, K. T. and Moulton, V. L., Some uses of the Farris transform in mathematics and phylogenetics – a review, Ann. Comb. 11(1): (2007), 1–37. doi:10.1007/s00026-007-0302-5.Google Scholar
Dress, A. W. M., Klucznik, M., Koolen, J. H. and Moulton, V. L., $2 kn-\binom{2k+1}{2}$: A note on extremal combinatorics of cyclic split systems, Sém. Lothar. Combin. 47 (2001), B47, pp. 17. https://eudml.org/doc/121892.Google Scholar
Dress, A. W. M. and Wu, Y., Phylogenetic analysis and mod 2 homology, Forum Math. 24(3): (2012), 495–505. doi:10.1515/form.2011.070.CrossRefGoogle Scholar
Dunwoody, M. J. and Krön, B., Vertex cuts, J. Graph Theory 80(2): (2015), 136–171. doi:10.1002/jgt.21844.CrossRefGoogle Scholar
Elbracht, C., Kneip, J. and Teegen, M., Trees of tangles in infinite separation systems, Math. Proc. Cambridge Philos. Soc. 173(2): (2022), 297–327. doi:10.1017/S0305004121000530.CrossRefGoogle Scholar
Evans, S. N., Probability and Real Trees, volume 1920 of Lecture Notes in Mathematics. (Springer, Berlin, 2005) doi:10.1007/978-3-540-74798-7.Google Scholar
Fischer, M., On the uniqueness of the maximum parsimony tree for data with up to two substitutions: An extension of the classic Buneman theorem in phylogenetics, Mol. Phylogenet. Evol 137 (2019), 127–137. doi:10.1016/j.ympev.2019.03.013.Google ScholarPubMed
Fischer, M., Defining binary phylogenetic trees using parsimony, Ann. Comb. 27(3): (2023), 457–467. doi:10.1007/s00026-022-00627-x.CrossRefGoogle Scholar
Fischer, M., van Iersel, L., Kelk, S. and Scornavacca, C., On computing the maximum parsimony score of a phylogenetic network, SIAM J. Discrete Math. 29(1): (2015), 559–585. doi:10.1137/140959948.CrossRefGoogle Scholar
Forcey, S., Circular planar electrical networks, split systems, and phylogenetic networks, SIAM J. Appl. Algebra Geom. 7(1): (2023), 49–76. doi:10.1137/22M1473844.CrossRefGoogle Scholar
Francis, A., Huber, K. T., Moulton, V. and Wu, T., Encoding and ordering $X$-cactuses, Adv. in Appl. Math. 142 (2023), 102414, pp. 31. doi:10.1016/j.aam.2022.102414. 31 pp.Google Scholar
Galashin, P. and Postnikov, A., Purity and separation for oriented matroids, Mem. Amer. Math. Soc. 289(1439): (2023) pp. v+79. doi:10.1090/memo/1439.Google Scholar
Gambette, P., Huber, K. T. and Scholz, G. E., Uprooted phylogenetic networks, Bull. Math. Biol. 79(9): (2017), 2022–2048. doi:10.1007/s11538-017-0318-x.Google ScholarPubMed
Gioan, E. and Paul, C., Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs, Discrete Appl. Math. 160(6): (2012), 708–733. doi:10.1016/j.dam.2011.05.007.Google Scholar
Goloboff, P. A. and Wilkinson, M., On defining a unique phylogenetic tree with homoplastic characters, Mol. Phylogenet. Evol 122 (2018), 95–101. doi:10.1016/j.ympev.2018.01.020.Google ScholarPubMed
Hellmuth, M., Huber, K. T., Moulton, V., Scholz, G. E. and Stadler, P. F., Injective split systems, Graphs Combin. 39(4): (2023), 65, pp. 22. doi:10.1007/s00373-023-02660-w.CrossRefGoogle Scholar
Hellmuth, M., Schaller, D. and Stadler, P. F., Compatibility of partitions with trees, hierarchies, and split systems, Discrete Appl. Math. 314 (2022), 265–283. doi:10.1016/j.dam.2022.03.014.CrossRefGoogle Scholar
Hellmuth, M., Schaller, D. and Stadler, P. F., Clustering systems of phylogenetic networks, Theory Biosci. 142(4): (2023), 301–358. doi:10.1007/s12064-023-00398-w.CrossRefGoogle ScholarPubMed
Helminck, P. A., Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves, Math. Z. 301(2): (2022), 1259–1297. doi:10.1007/s00209-021-02959-5.CrossRefGoogle Scholar
Hendriksen, M. and Kapust, N., On the comparison of incompatibility of split systems across different numbers of taxa, Bull. Math. Biol. 83(7): (2021), 78, pp. 15. doi:10.1007/s11538-021-00911-6.CrossRefGoogle ScholarPubMed
Huber, K. T., Moulton, V., Semple, C. and Wu, T., Representing partitions on trees, SIAM J. Discrete Math. 28(3): (2014), 1152–1172. doi:10.1137/130906192.CrossRefGoogle Scholar
Kleinman, A., Harel, M. and Pachter, L., Affine and projective tree metric theorems, Ann. Comb. 17(1): (2013), 205–228. doi:10.1007/s00026-012-0173-2.CrossRefGoogle Scholar
Koloschin, M., Krill, T. and Pitz, M., End spaces and tree-decompositions, J. Combin. Theory Ser. B 161 (2023), 147–179. doi:10.1016/j.jctb.2023.02.007.CrossRefGoogle Scholar
Kupavskii, A., Pach, J. and Tomon, I., On the size of K-cross-free families, Combinatorica 39(1): (2019), 153–164. doi:10.1007/s00493-017-3792-8.CrossRefGoogle Scholar
Li, S-D., Fan, Y-Z., Ito, T., Karimi, M. and Xu, J., The isomorphism problem of trees from the viewpoint of Terwilliger algebras, Id/No 177: (2021), 105328, pp. 11. doi:10.1016/j.jcta.2020.105328.Google Scholar
Liu, P., A tree distinguishing polynomial, Discrete Appl. Math. 288 (2021), 1–8. doi:10.1016/j.dam.2020.08.019.Google Scholar
Liu, P., Biller, P., Gould, M. and Colijn, C., Analyzing phylogenetic trees with a tree lattice coordinate system and a graph polynomial, Systematic Biology 71(6): (2022), 1378–1390. doi:10.1093/sysbio/syac008.CrossRefGoogle Scholar
Loehr, N. A. and Warrington, G. S., A rooted variant of Stanley’s chromatic symmetric function, Discrete Math. 347(3): (2024), 113805, pp. 18. doi:10.1016/j.disc.2023.113805.CrossRefGoogle Scholar
Misra, N., Blelloch, G., Ravi, R. and Schwartz, R., Generalized Buneman pruning for inferring the most parsimonious multi-state phylogeny, J. Comput. Biol. 18(3): (2011), 445–457. doi:10.1089/cmb.2010.0254.CrossRefGoogle ScholarPubMed
Orellana, R. and Scott, G., Graphs with equal chromatic symmetric functions, Discrete Math. 320 (2014), 1–14. doi:10.1016/j.disc.2013.12.006.CrossRefGoogle Scholar
Oum, S., Rank-width and vertex-minors, J. Combin. Theory Ser. B 95(1): (2005), 79–100. doi:10.1016/j.jctb.2005.03.003.CrossRefGoogle Scholar
Petrov, E., Weak similarities of finite ultrametric and semimetric spaces, p-Adic Numbers Ultrametric Anal. Appl. 10(2): (2018), 108–117. doi:10.1134/S2070046618020048.CrossRefGoogle Scholar
Pons, J. C., Coronado, T. M., Hendriksen, M. and Francis, A., A polynomial invariant for a new class of phylogenetic networks, PLoS One 17(5): (2022), e0268181, pp. 22. doi:10.1371/journal.pone.0268181.CrossRefGoogle ScholarPubMed
Qian, C., Wu, Y. and Xie, M.. How to tell a phylogenetic tree from its 2-split system?, 2025.Google Scholar
Qian, C., Wu, Y. and Xie, M.. A phylogenetic tree with at least $2\ell+5$ leaves is determined by its score-$\leq \ell$ split system, 2025. URL: https://math.sjtu.edu.cn/faculty/ykwu/data/Paper/split202507.pdf.Google Scholar
Qian, C., Wu, Y. and Xie, M.. Reconstruct a phylogenetic tree with at least $2\ell+3$ leaves from its score-$\ell$ split system, 2025.Google Scholar
Schröter, B., Multi-splits and tropical linear spaces from nested matroids, Discrete Comput. Geom. 61(3): (2019), 661–685. doi:10.1007/s00454-018-0021-1.CrossRefGoogle Scholar
Semple, C. and Steel, M., Phylogenetics. Oxford Lecture Series in Mathematics and its Applications., Volume 24, (Oxford University Press, Oxford, 2003).Google Scholar
Steel, M.. Phylogeny – Discrete and Random Processes in Evolution, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), 2016, Volume 89 Philadelphia, PA, doi:10.1137/1.9781611974485.ch1.CrossRefGoogle Scholar
Tserunyan, A., A descriptive construction of trees and Stallings’ theorem. In Trends in Set Theory, volume of Contemp. Math. Volume 752, pp. 191–207 (Amer. Math. Soc. Providence, RI, 2020) doi:10.1090/conm/752/15137.CrossRefGoogle Scholar
Wilde, M. and Fischer, M., Defining binary phylogenetic trees using parsimony: new bounds, Discrete Appl. Math. 341 (2023), 257–269. doi:10.1016/j.dam.2023.08.005.CrossRefGoogle Scholar
Wilde, M. and Fischer, M., A complete characterization of pairs of binary phylogenetic trees with identical Ak-alignments. 2024, arXiv:2408.07011.Google Scholar
Wilkes, G. R.. On the structure of vertex cuts separating the ends of a graph, Pacific J. Math. 278(2): (2015), 479–510. doi:10.2140/pjm.2015.278.479.Google Scholar
Wu, Y., Lit-only sigma game on a line graph, European J. Combin. 30(1): (2009), 84–95. doi:10.1016/j.ejc.2008.02.003.Google Scholar