A generalized Robinson-Foulds distance for labeled trees

Abstract

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.

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

“…The Robinson-Foulds (RF) distance is defined in the literature for rooted and unrooted trees. Moreover, as mentioned in [5], the problem of computing the RF distance for two rooted trees can be reduced to computing the RF distance for the two corresponding unrooted trees obtained by grafting an edge linking the root to a dummy leaf. Therefore, in this paper we restrict ourselves to unrooted trees.…”

“…A previous extension of RF to labeled trees, based on edit operations on edges rather than on nodes, was introduced in [5]. This distance, which we call ELRF , was defined on three operations: Edge extension Ext ( T, x, X ) creating an edge { x, y } and defined as a node insertion Ins ( T, y, x, X, λ ( x )) inserting a node y as a neighbour of x and assigning to y the label of x ; Edge contraction Cont ( T, { x, y }) is equal to a node deletion Del ( T, y, x ) deleting y , but contrary to LRF, requires that λ ( x ) = λ ( y ); Node flip Flip ( T, x, λ ) assigning the label λ to x . …”

“…While a good edge e in T has a corresponding good edge e ′ in T ′ (the one defining the same bipartition), a bad edge in T has no corresponding edge in T ′. However, these bad edges may be grouped into pairs of corresponding islands (called maximum bad subtrees in [5]), as defined bellow.…”

“…The metric we developed in [5], referred to as ELRF , is the first effort towards comparing labeled gene trees, expressed in terms of trees with a binary node labeling (typically speciation and duplication). ELRF is an extension of the RF distance, one of the most widely used tree distance, not only in phylogenetics, but also in other fields such as in linguistics, for its computational efficiency, intuitive interpretation and the fact that it is a true metric.…”

“…Classically defined in terms of bipartition or clade dissimilarity, the RF distance can similarly be defined in terms of edit operations on tree edges: the minimum number of edge contraction and extension needed to transform one tree into the other [22]. In [5], this definition of the RF distance was extended to trees with binary node labeling by including a node flip operation, alongside edge contractions and extensions. While remaining a metric, ELRF turned out to be much more challenging to compute.…”