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

A large variety of pairwise measures of similarity or dissimilarity have been developed for comparing phylogenetic trees, for example, species trees or gene trees. Due to its intuitive definition in terms of tree clades and bipartitions and its computational efficiency, the Robinson-Foulds (RF) distance is the most widely used for trees with unweighted edges and labels restricted to leaves (representing the genetic elements being compared). However, in the case of gene trees, an important information revealing the nature of the homologous relation between gene pairs (orthologs, paralogs, and xenologs) is the type of event associated to each internal node of the tree, typically speciations or duplications, but other types of events may also be considered, such as horizontal gene transfers. This labeling of internal nodes is usually inferred from a gene tree/species tree reconciliation method. Here, we address the problem of comparing such event-labeled trees. The problem differs from the classical problem of comparing uniformly labeled trees (all labels belonging to the same alphabet) that may be done using the Tree Edit Distance (TED) mainly due to the fact that, in our case, two different alphabets are considered for the leaves and internal nodes of the tree, and leaves are not affected by edit operations. We propose an extension of the RF distance to event-labeled trees, based on edit operations comparable to those considered for TED: node insertion, node deletion, and label substitution. We show that this new Labeled Robinson-Foulds (LRF) distance can be computed in linear time, in addition of maintaining other desirable properties: being a metric, reducing to RF for trees with no labels on internal nodes and maintaining an intuitive interpretation. The algorithm for computing the LRF distance enables novel analyses on event-label trees such as reconciled gene trees. Here, we use it to study the impact of taxon sampling on labeled gene tree inference and conclude that denser taxon sampling yields trees with better topology but worse labeling. [Algorithms; combinatorics; gene trees; phylogenetics; Robinson-Foulds; tree distance.].

A generalized Robinson-Foulds distance for labeled trees.

BACKGROUND: The Robinson-Foulds (RF) distance is a well-established measure between phylogenetic trees. Despite a lack of biological justification, it has the advantages of being a proper metric and being computable in linear time. For phylogenetic applications involving genes, however, a crucial aspect of the trees ignored by the RF metric is the type of the branching event (e.g. speciation, duplication, transfer, etc). RESULTS: We extend RF to trees with labeled internal nodes by including a node flip operation, alongside edge contractions and extensions. We explore properties of this extended RF distance in the case of a binary labeling. In particular, we show that contrary to the unlabeled case, an optimal edit path may require contracting "good" edges, i.e. edges shared between the two trees. CONCLUSIONS: We provide a 2-approximation algorithm which is shown to perform well empirically. Looking ahead, computing distances between labeled trees opens up a variety of new algorithmic directions.Implementation and simulations available at https://github.com/DessimozLab/pylabeledrf .