Abstract
Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any ‘triplet cover’ of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are ‘shellable’ and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths.
Highlights
Trees play a central role in systematic biology, and other areas of classification, such as linguistics
We show that in that case the resulting triplet cover is ‘shellable’ which implies that the inter-leaf distances defined on these pairs uniquely determine the tree and its edge lengths
This corollary has two important implications. First it implies [from results in Dress et al (2012)] that if T is a minimum triplet cover for T, T can be uniquely reconstructed from the tree metric restricted to the pairs in
Summary
Trees play a central role in systematic biology, and other areas of classification, such as linguistics. We do not require distance values for all of the n 2 pairs from X (where n = |X |), since just 2n − 3 carefully selected pairs of leaves suffice to determine T and its edge lengths [see Guénoche et al (2004); more recent results appear in Dress et al (2012), motivated by the irregular distribution of genes across species in biological data]. This value of 2n − 3 cannot be made any smaller, since a binary unrooted tree with n leaves has 2n − 3 edges, and the inter-leaf distances are linear combinations of the corresponding 2n − 3 edge lengths (so, by linear algebra, these values cannot be uniquely determined by fewer than 2n − 3 equations).
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