Abstract
It is a classical result that an unrooted tree $T$ having positive realvalued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each nonleaf vertex of $T$ has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in $T$ is the following: for each nonleaf vertex $v$, choose a leaf in each of the three components of $Tv$, group these three leaves into three pairs, and take the union of this set over all choices of $v$. This forms a socalled `triplet cover' for $T$. In the first part of this paper we answer an open question (from 2012) by showing that the induced leaftoleaf distances for any triplet cover for $T$ uniquely determine $T$ and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, `sparse', and `shellable'.
Original language  English 

Pages (fromto)  5982 
Number of pages  24 
Journal  Advances in Applied Mathematics 
Volume  99 
Early online date  24 Apr 2018 
DOIs  
Publication status  Published  Aug 2018 
Keywords
 Phylogenetic tree
 triplet cover
 treedistances
 Hall's theorem
 ample patchwork
 shellability
Profiles

Katharina Huber
 School of Computing Sciences  Associate Professor
 Computational Biology  Member
Person: Research Group Member, Academic, Teaching & Research

Vincent Moulton
 School of Computing Sciences  Professor in Computational Biology
 Norwich Epidemiology Centre  Member
 Computational Biology  Member
Person: Research Group Member, Academic, Teaching & Research