Are node-based and stem-based clades equivalent? Insights from graph theory

Despite the prominence of “tree-thinking” among contemporary systematists and evolutionary biologists, the biological meaning of different mathematical representations of phylogenies may still be muddled. We compare two basic kinds of discrete mathematical models used to portray phylogenetic relationships among species and higher taxa: stem-based trees and node-based trees. Each model is a tree in the sense that is commonly used in mathematics; the difference between them lies in the biological interpretation of their vertices and edges. Stem-based and node-based trees carry exactly the same information and the biological interpretation of each is similar. Translation between these two kinds of trees can be accomplished by a simple algorithm, which we provide. With the mathematical representation of stem-based and node-based trees clarified, we argue for a distinction between types of trees and types of names. Node-based and stem-based trees contain exactly the same information for naming clades. However, evolutionary concepts, such as monophyly, are represented as different mathematical substructures in the two models. For a given stem-based tree, one should employ stem-based names, whereas for a given node-based tree, one should use node-based names, but applying a node-based name to a stem-based tree is not logical because node-based names cannot exist on a stem-based tree and visa versa. Authors might use node-based and stem-based concepts of monophyly for the same representation of a phylogeny, yet, if so, they must recognize that such a representation differs from the graphical models used for computing in phylogenetic systematics.