Bulletin of Mathematical Biology

, Volume 80, Issue 6, pp 1563–1577 | Cite as

Reconstructing Unrooted Phylogenetic Trees from Symbolic Ternary Metrics

  • Stefan Grünewald
  • Yangjing Long
  • Yaokun Wu
Original Article


Böcker and Dress (Adv Math 138:105–125, 1998) presented a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree T with leaf set X and a proper vertex coloring of its interior vertices, we can map every triple of three different leaves to the color of its median vertex. We characterize all ternary maps that can be obtained in this way in terms of 4- and 5-point conditions, and we show that the corresponding tree and its coloring can be reconstructed from a ternary map that satisfies those conditions. Further, we give an additional condition that characterizes whether the tree is binary, and we describe an algorithm that reconstructs general trees in a bottom-up fashion.


Symbolic ternary metric Median vertex Unrooted phylogenetic tree 



We thank the anonymous referees for their helpful comments and suggestions. We thank Peter F. Stadler for suggesting to consider general median graphs and Zeying Xu for some useful comments. This work is supported by the NSFC (11671258) and STCSM (17690740800). YL acknowledges support of Postdoctoral Science Foundation of China (No. 2016M601576).


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.CAS-MPG Partner Institute for Computational BiologyChinese Academy of Sciences Key Laboratory of Computational BiologyShanghaiChina
  2. 2.School of Mathematics and StatisticsCentral China Normal UniversityWuhanChina
  3. 3.Department of Mathematics and MOE-LSCShanghai Jiao Tong UniversityShanghaiChina

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