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

Foundations of the wald space for phylogenetic trees

Lueg, J., Garba, M. K., Nye, T. M. W., & Huckemann, S. (2024) · Journal of the London Mathematical Society · doi:10.1112/jlms.12893 · Cited by 3

Tropical geometric tools for machine learning: the TML package

Barnhill, D., Yoshida, R., Aliatimis, G., & Miura, K. (2024) · Journal of Software for Algebra and Geometry · doi:10.2140/jsag.2024.14.133 · Cited by 3

A characterization of dissimilarity families of trees

Baldisserri, A., & Rubei, E. (2017) · Discrete Applied Mathematics · doi:10.1016/j.dam.2016.12.007 · Cited by 3

On Urysohn’s ℝ-Tree

Berestovskii, V. N. (2019) · Siberian Mathematical Journal · doi:10.1134/s0037446619010026 · Cited by 3

A Characterization of Minimum Spanning Tree-Like Metric Spaces

Hayamizu, M., Endô, H., & Fukumizu, K. (2016) · IEEE Transactions on Computational Biology and Bioinformatics · doi:10.1109/tcbb.2016.2550431 · Cited by 3

Imputing Phylogenetic Trees Using Tropical Polytopes over the Space of Phylogenetic Trees

Yoshida, R. (2023) · Mathematics · doi:10.3390/math11153419 · Cited by 5

Multitriangulations and Tropical Pfaffians

Ruiz, L. C., & Santos, F. (2024) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1527507 · Cited by 5

A Ping Too Far: Real World Network Latency Measurement

Jackson, G., Keleher, P. J., & Sussman, A. (2015) · IEEE International Conference on eScience · doi:10.1109/escience.2015.74 · Cited by 3

The dual complex of $${\overline{M}_{0,n}}$$ M ¯ 0 , n via phylogenetics

Giansiracusa, N. (2016) · Archiv der Mathematik · doi:10.1007/s00013-016-0910-6 · Cited by 3

Shortest-Paths Trees

Wu, B. Y., & Chao, K.-M. (2004) · Chapman & Hall/CRC eBooks · doi:10.1201/9780203497289-9 · Cited by 3

Expansion of gene clusters, circular orders, and the shortest Hamiltonian path problem

2017, T. S. O. T. G. A. B. N. L., Prohaska, S. J., Berkemer, S. J., Gärtner, F., Gatter, T., Retzlaff, N., zu Siederdissen, C. H., & Stadler, P. F. (2017) · Journal of Mathematical Biology · doi:10.1007/s00285-017-1197-3 · Cited by 3

Bio-inspired Knowledge Representation Framework for Decision Making in Product Design

Tiwari, V., Jain, P. K., & Tandon, P. (2019) · Lecture notes on multidisciplinary industrial engineering · doi:10.1007/978-981-32-9417-2_48 · Cited by 3

An algorithm for finding a representation of a subtree distance

Ando, K., & Sato, K. (2017) · Journal of Combinatorial Optimization · doi:10.1007/s10878-017-0145-x · Cited by 3

Representations of graphs and networks (coding, layouts and embeddings)

Kozyrev, V. P., & Yushmanov, S. V. (1992) · Journal of Mathematical Sciences · doi:10.1007/bf01097528 · Cited by 3

A note on convexity and submatrices of distance matrices

Simões-Pereira, J. M. S. (1987) · Linear and Multilinear Algebra · doi:10.1080/03081088708817769 · Cited by 3

Unrooted trees revisited: Topology and poetic data

Juillard, M., & Luong, N. X. (1989) · Computers and the Humanities · doi:10.1007/bf00056144 · Cited by 3

An Investigation into Graph Curvature's Ability to Measure Congestion in Network FlowPreprint

Yancey, M. (2015) · arXiv (Cornell University) · Cited by 3

Multi-State Perfect Phylogeny Mixture Deconvolution and Applications to Cancer SequencingPreprint

El-Kebir, M., Satas, G., Oesper, L., & Raphael, B. J. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1604.02605 · Cited by 3

Modelling complex population structure using F -statistics and Principal Component AnalysisPreprint

Peter, B. M. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.07.13.452141 · Cited by 3

A practical study of the proportion of non-unique neighbor-joining trees of microsatellite markers

Segura-Alabart, N., Serratosa, F., & Fernández, A. (2024) · Computational and Structural Biotechnology Reports · doi:10.1016/j.csbr.2024.100022 · Cited by 2

Extension Results for Lipschitz Mappings

Cobzaş, Ş., Miculescu, R., & Nicolae, A. (2019) · Lecture notes in mathematics · doi:10.1007/978-3-030-16489-8_4 · Cited by 2

Determining triangulations and quadrangulations by boundary distances

Haslegrave, J. (2023) · Journal of Combinatorial Theory Series B · doi:10.1016/j.jctb.2023.08.002 · Cited by 2

Tropical Neural Networks and Its Applications to Classifying Phylogenetic Trees

Yoshida, R., Aliatimis, G., & Miura, K. (2024) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn60899.2024.10650971 · Cited by 2

Optimizing over path-length matrices of unrooted binary trees

Catanzaro, D., Pesenti, R., Sapucaia, A., & Wolsey, L. A. (2025) · Mathematical Programming · doi:10.1007/s10107-025-02226-z · Cited by 2

Scalable Algorithms for Large and Dynamic Networks: Reducing Big Data for Small Computations

Saniee, I. (2015) · Bell Labs Technical Journal · doi:10.15325/bltj.2015.2437465 · Cited by 2

Representation of a tree with p hanging vertices by 2p?3 elements of its distance matrix

Yushmanov, S. V. (1984) · Mathematical Notes · doi:10.1007/bf01139951 · Cited by 2

Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

Evans, S. N., & Lanoue, D. (2018) · Advances in Applied Mathematics · doi:10.1016/j.aam.2018.01.001 · Cited by 2

Composed Degree-Distance Realizations of Graphs

Bar-Noy, A., Peleg, D., Perry, M., & Rawitz, D. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-79987-8_5 · Cited by 2

Quasisymmetric mappings in b-metric spaces

Petrov, E., & Salimov, R. R. (2021) · Ukrainian Mathematical Bulletin · doi:10.37069/1810-3200-2021-18-1-4 · Cited by 2

connected Joins in Graphs

Sebö, A., & Tannier, É. (2001) · Lecture notes in computer science · doi:10.1007/3-540-45535-3_30 · Cited by 2

The geometry of independence tree models with hidden variables

Zwiernik, P., & Smith, J. Q. (2010) · Warwick Research Archive Portal (University of Warwick) · doi:10.82444/warw.33597640 · Cited by 2

Underlying graph and total length of optimal realizations of variable distance matrices

Simões-Pereira, J. M. S. (1987) · Graphs and Combinatorics · doi:10.1007/bf01788561 · Cited by 2

The Cavender-Farris-Neyman Model with a Molecular ClockPreprint

Coons, J. I., & Sullivant, S. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.04175 · Cited by 2

Projection on an Acute Symmetrical Simplicial Closed Convex Cone and its Application to Star Graphs

Fichet, B. (1986) · COMPSTAT · doi:10.1007/978-3-642-46890-2_26 · Cited by 2

A note on generalized four-point inequality

Petrov, E., & Salimov, R. R. (2023) · Journal of Mathematical Sciences · doi:10.1007/s10958-023-06507-w · Cited by 1

Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

Ilmavirta, J., Lassas, M. J., Lu, J., Oksanen, L., & Ylinen, L. (2024) · Inverse Problems and Imaging · doi:10.3934/ipi.2024049 · Cited by 1

Topological data analysis captures horizontal gene transfer in antimicrobial resistance gene families among clinically relevant bacteria

Guerrero‐Flores, S., Contreras-Peruyero, H., Ibarra-Rodríguez, J. M., Lovaco-Flores, J. A., la Rosa, F. S. N.-D., Fontove-Herrera, F., & Mojica, N. S. (2025) · Frontiers in Microbiology · doi:10.3389/fmicb.2025.1461293 · Cited by 1

New heuristics for phylogeny estimation under the balanced minimum evolution criterion

Catanzaro, D., Dehaybe, H., & Pesenti, R. (2025) · Bioinformatics · doi:10.1093/bioinformatics/btaf361 · Cited by 1

On the weights of simple paths in weighted complete graphs

Rubei, E. (2013) · Indian Journal of Pure and Applied Mathematics · doi:10.1007/s13226-013-0027-6 · Cited by 1

IDXL: Species Tree Inference Using Internode Distance and Excess Gene Leaf Count

Bhattacharyya, S., & Mukherjee, J. (2017) · Journal of Molecular Evolution · doi:10.1007/s00239-017-9807-7 · Cited by 1

Betweenness structures of small linear co-size

Szabó, P. G. N. (2022) · Discrete Applied Mathematics · doi:10.1016/j.dam.2022.08.023 · Cited by 1

Ultrametrics and Surface Singularities

Popescu‐Pampu, P. (2020) · Lecture notes in mathematics · doi:10.1007/978-3-030-61807-0_9 · Cited by 1

HyperAid

Chien, E., Tabaghi, P., & Milenković, O. (2022) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/3534678.3539378 · Cited by 1

A massively parallel branch-&-bound algorithm for the balanced minimum evolution problem

Catanzaro, D., Frohn, M., Gascuel, O., & Pesenti, R. (2023) · Computers & Operations Research · doi:10.1016/j.cor.2023.106308 · Cited by 1

The maximum four point condition matrix of a tree

Azimi, A., Jana, R., Nagar, M. K., & Sivasubramanian, S. (2024) · Linear Algebra and its Applications · doi:10.1016/j.laa.2024.03.023 · Cited by 1

Boundary rigidity of CAT(0) cube complexes

Chalopin, J., & Chepoi, V. D. (2024) · Journal of Combinatorial Theory Series B · doi:10.1016/j.jctb.2024.07.003 · Cited by 1

New generalized metric based on branch length distance to compare B cell lineage trees

Farnia, M., & Tahiri, N. (2024) · Algorithms for Molecular Biology · doi:10.1186/s13015-024-00267-1 · Cited by 1

A combinatorial formula for principal minors of a matrix with tree-metric exponents and its applications

Hirai, H., & Yabe, A. (2015) · Journal of Combinatorial Theory Series A · doi:10.1016/j.jcta.2015.02.005 · Cited by 1

Constructing phylogenetic networks from trees

Bereg, S., & Bean, K. (2005) · IEEE International Conference on Bioinformatics and Bioengineering (BIBE) · doi:10.1109/bibe.2005.19 · Cited by 1

Rank of a matrix of block graphs

Chen, Y., & Liao, Y. (2021) · International Journal of Quantum Chemistry · doi:10.1002/qua.26748 · Cited by 1

On tight spans and tropical polytopes for directed distancesPreprint

Hirai, H., & Koichi, S. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1004.0415 · Cited by 1

An Optimality Criterion and the Total Length of the Graph Realization of a Distance Matrixa

Simões-Pereira, J. M. S. (1989) · Annals of the New York Academy of Sciences · doi:10.1111/j.1749-6632.1989.tb22476.x · Cited by 1

Decentralized Network Bandwidth Prediction and Node Search

Song, S. (2011) · IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum · doi:10.1109/ipdps.2011.369 · Cited by 1

20. Theorietag der GI-Fachgruppe "Automaten und Formale Sprachen''

Damm, C., Holzer, M., Kutrib, M., Leupold, P., Plátek, M., Lopatková, M., Vollmer, H., Alhazov, A., Freund, R., Morita, K., et al. (2010) · Kobra (Universitätsbibliothek Kassel) · Cited by 1

Classifications of ultrametric spaces according to roundnessPreprint

Faver, T. E., Kochalski, K., Murugan, M., Verheggen, H., Wesson, E. N., & Weston, A. (2012) · arXiv (Cornell University) · Cited by 1

Gromov meets Phylogenetics - new Animals for the Zoo of Biocomputable Metrics on Tree SpacePreprint

Liebscher, V. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1504.05795 · Cited by 1

Graphlike families of multiweightsPreprint

Baldisserri, A., & Rubei, E. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.09183 · Cited by 1

Modeling the distribution of distance data in Euclidean space

Davidson, R., Rusinko, J., Vernon, Z., & Xi, J. (2017) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/685/13750 · Cited by 1

Gaussian latent tree model constraints for linguistics and other applications

Shiers, N. (2026) · Warwick Research Archive Portal (University of Warwick) · doi:10.82444/warw.33724468 · Cited by 1

A characterization of distance matrices of weighted cubic graphs and Peterson graphsPreprint

Rubei, E., & Ziani, D. V. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1901.00360 · Cited by 1

Sets of double and triple weights of treesPreprint

Rubei, E. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0712.3026 · Cited by 1

Faithful tropicalization of the Grassmannian of planesPreprint

Cueto, M. A., Häbich, M., & Werner, A. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1309.0450 · Cited by 1

On graphlike k-dissimilarity vectorsPreprint

Baldisserri, A., & Rubei, E. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1211.0423 · Cited by 1

Invariants Recovering the Reduction Type of a Hyperelliptic CurvePreprint

Kellock, L. C., & García, E. L. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5204576 · Cited by 1

The Median Extension of Data Analysis Metric Structures

Benayade, M., & Fichet, B. (2002) · Birkhäuser Basel eBooks · doi:10.1007/978-3-0348-8201-9_30 · Cited by 1

Un panorama des approximations en norme du supremum pour la classification

Fichet, B., & Seston, M. (2010) · Mathématiques et sciences humaines · doi:10.4000/msh.11746

Phylogenetic Trees From Sequences

Ryvkin, P., & Wang, L. (2010) · Springer eBooks · doi:10.1007/978-0-387-09760-2_6

On dissimilarity vectors of (not necessarily positive) weighted treesPreprint

Rubei, E. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1004.3406

Hyperconvexity and Metric TreesPreprint

Aksoy, A. G., & Maurizi, B. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0702125

Finding Centers and Medians of a Tree by Distance Queries

Wu, B. Y. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-07890-8_30

Nonultrametric triangles in diametral additive metric spaces

Faver, T. E., Kochalski, K., Murugan, M., Verheggen, H., Wesson, E. N., & Weston, A. (2015) · Involve a Journal of Mathematics · doi:10.2140/involve.2015.8.33

Compact Widts in Metric TreesPreprint

Aksoy, A. G., & Kinneberg, K. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1007.2208

A convergence result on random products of mappings in metric spaces

Khamsi, M. A., & Louhichi, I. (2012) · Fixed Point Theory and Applications · doi:10.1186/1687-1812-2012-43

Hereditarily Optimal Realizations

Lesser, A. (2004)

A note on tree realizations of matrices

Hertz, A., & Varone, S. C. (2007) · RAIRO - Operations Research · doi:10.1051/ro:2007028

Constructing Phylogenetic Trees from Subsplits

Kashiwada, A. (2005) · Scholarship - Claremont (Claremont Colleges)

A priori assessment of data quality in molecular

Meusemann, K., von Reumont, B. M., Kück, P., & Prohaska, S. J. (2014)

Topics in abstract order geometry

Retter, W. (2014) · tub.dok (Hamburg University of Technology) · doi:10.15480/882.1154

Moduli Space of Families of Positive (n − 1)-Weights

Calamai, S. (2019) · Journal of Classification · doi:10.1007/s00357-019-9305-2

Tropical Grassmannian and Tropical Linear Varieties from phylogenetic treesPreprint

Dukkipati, A., & Sen, A. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1312.0752

The dual complex of $\bar{M}_{0,n}$ via phylogeneticsPreprint

Giansiracusa, N. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1512.00323

Inferenza filogenetica con il metodo della minima evoluzione bilanciata

Zaccaria, F. (2013)

Families of $2$-weights of some particular graphsPreprint

Baldisserri, A., & Rubei, E. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1605.00946

Alignment and Scoring

Koski, T. (2001) · Computational biology · doi:10.1007/978-94-010-0612-5_5

Dissimilarity Vectors of Trees and Their Tropical Linear Spaces (Extended Abstract)

Giraldo, B. I. (2011) · Discrete Mathematics & Theoretical Computer Science · doi:10.46298/dmtcs.2918

An Algorithm for Finding a Representation of a Subtree Distance

Ando, K., & Sato, K. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-45587-7_22

A characterization of uniquely representable graphs

Szabó, P. G. N. (2021) · Discussiones Mathematicae Graph Theory · doi:10.7151/dmgt.2415

Algèbre de groupe en caractéristique 1 et distances invariantes sur un groupe fini

Castella, D., & Gaubert, S. (2017) · Mathematische Zeitschrift · doi:10.1007/s00209-017-1971-3

Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb R$-treesPreprint

Aksoy, A. G., Al-Ansari, M., & Peng, Q. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1601.03108

The tower matrix, an alternative to deal with distances and quasi-distances

Zhang, Y., da Costa, A. V., & Simões-Pereira, J. M. S. (2019) · Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)

On the weights of simple paths in weighted complete graphsPreprint

Rubei, E. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1210.0845

Roundness properties of ultrametric spacesPreprint

Faver, T. E., Kochalski, K., Murugan, M., Verheggen, H., Wesson, E. N., & Weston, A. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1201.6669

Weighted graphs with distances in given rangesPreprint

Rubei, E. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1409.3863

Compact widths in metric trees

Aksoy, A. G., & Kinneberg, K. (2011) · Banach Center Publications · doi:10.4064/bc92-0-1

Anti-consensus: detecting trees that have an evolutionary signal that is lost in consensusPreprint

Huson, D. H., Albrecht, B., Patz, S., & Steel, M. (2019) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/706416

Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree SpacePreprint

Lee, W., Li, W., Lin, B., & Monod, A. (2019) · arXiv (Cornell University)

Kantorovich distance on a finite metric spacePreprint

Montrucchio, L., & Pistone, G. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.07547

Negatively Curved GraphsPreprint

Yancey, M. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1512.01281

Realization of distance matrices by graphs of genus 1Preprint

Bocci, C., & Capresi, C. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.07698

Bibliography

Wheeler, W. C. (2012) · SYSTEMATICS · doi:10.1002/9781118301081.biblio

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