Defining phylogenetic relationships: a complete and minimal set of combinatorial properties for triplets

Abstract

Triplets, as minimal informative rooted trees, are fundamental units of information in phylogenetics. Their importance for phylogenetic reconstruction, cladistic biogeography, or supertree methods relies on the fact that any rooted tree can be decomposed into a set of triplets. In order to formalize the tree building from a consistent triplet set, several k -adic rules of inference, i.e., rules that allow us to deduce at least one new triplet from exactly k other ones, have been identified. However, it remains unclear whether it is possible to reduce all the possible k -adic rules to a finite set of basic properties. In order to solve this problem, we propose here to define triplets in terms of degree of equivalence relations. Given the axiomatic definition of the latter, we establish a list of the most basic properties for triplets. With such an approach, we finally prove that the closure of any coherent triplet set can be computed uniquely from these basic properties.