Species notions that combine phylogenetic trees and phenotypic partitions

A recent paper (Manceau and Lambert, 2016) developed a novel approach for describing two well-defined notions of 'species' based on a phylogenetic tree and a phenotypic partition. In this paper, we explore some further combinatorial properties of this approach and describe an extension that allows an arbitrary number of phenotypic partitions to be combined with a phylogenetic tree for these two species notions.