A Linear Time Solution to the Labeled Robinson-Foulds Distance Problem

Motivation Comparing trees is a basic task for many purposes, and especially in phylogeny where different tree reconstruction tools may lead to different trees, likely representing contradictory evolutionary information. While a large variety of pairwise measures of similarity or dissimilarity have been developed for comparing trees with no information on internal nodes, very few address the case of inner node-labeled trees. Yet such trees are common; for instance reconciled gene trees have inner nodes labeled with the type of event giving rise to them, typically speciation or duplication. Recently, we proposed a formulation of the Labeled Robinson Foulds edit distance with edge extensions, edge contractions between identically labeled nodes, and node label flips. However, this distance proved difficult to compute, in particular because shortest edit paths can require contracting “good” edges, i.e. edges present in the two trees.

Results Here, we report on a different formulation of the Labeled Robinson Foulds edit distance — based on node insertion, deletion and label substitution — which we show can be computed in linear time. The new formulation also maintains other desirable properties: being a metric, reducing to Robinson Foulds for unlabeled trees and maintaining an intuitive interpretation. The new distance is computable for an arbitrary number of label types, thus making it useful for applications involving not only speciations and duplications, but also horizontal gene transfers and further events associated with the internal nodes of the tree. To illustrate the utility of the new distance, we use it to study the impact of taxon sampling on labeled gene tree inference, and conclude that denser taxon sampling yields better trees.