Encoding phylogenetic trees in terms of weighted quartets
One of the main problems in phylogenetics is to develop systematic methods for constructing evolutionary or phylogenetic trees. For a set of species X, an edge-weighted phylogenetic X-tree or phylogenetic tree is a (graph theoretical) tree with leaf set X and no degree 2 vertices, to- gether with a map assigning a non-negative length to each edge of the tree. Within phylogenetics, several methods have been proposed for con- structing such trees that work by trying to piece together quartet trees on X, i.e. phylogenetic trees each having four leaves in X. Hence, it is of interest to characterise when a collection of quartet trees corresponds to a (unique) phylogenetic tree. Recently, Dress and Erd os provided such a characterisation for binary phylogenetic trees, that is, phylogenetic trees all of whose internal vertices have degree 3. Here we provide a new char- acterisation for arbitrary phylogenetic trees.