Can We “Future-Proof” Consensus Trees?†

Abstract

Consensus methods are widely used for combining phylogenetic trees into a single estimate of the evolutionary tree for a group of species. As more taxa are added, the new source trees may begin to tell a different evolutionary story when restricted to the original set of taxa. However, if the new trees, restricted to the original set of taxa, were to agree exactly with the earlier trees, then we might hope that their consensus would either agree with or resolve the original consensus tree. In this article, we ask under what conditions consensus methods exist that are "future proof" in this sense. While we show that some methods (e.g., Adams consensus) have this property for specific types of input, we also establish a rather surprising "no-go" theorem: there is no "reasonable" consensus method that satisfies the future-proofing property in general. We then investigate a second notion of "future proofing" for consensus methods, in which trees (rather than taxa) are added, and establish some positive and negative results. We end with some questions for future work.

Impossibility Results on Stability of Phylogenetic Consensus Methods

“…One of the main results of [6] is that no regular 2-consensus method is extension stable. In the same paper, the feasibility of different relaxations of extension stability was discussed.…”

“…In the same paper, the feasibility of different relaxations of extension stability was discussed. The first question left open in [6] is about extension stability under restriction of the domain of consensus methods to binary trees. More precisely, fix a set X of taxa and let RBP(X) ⊆ RP(X) denote the subset of all rooted binary phylogenetic trees -i.e., the phylogenetic trees on X where every internal vertex has exactly two children.…”

“…Remark 2.2 (Related definitions in the literature). Our use of the term "consensus method" conforms to [5,6], whereas the term "consensus function on X" matches [14]. However, terminology is not completely consistent throughout the literature: in particular, we remark on some instances where the objects we introduced in Definition 2.1 appear under different names.…”

“…We now recall some general axioms for consensus methods. We follow [6] and call a consensus method ϕ regular if it satisfies the following three axioms:…”

“…Aho consensus and Adams consensus are Pareto on rooted triples. As we will discuss later on, strict consensus is 4 associatively stable (see Definition 3.4), whereas majority consensus, loose consensus, Aho consensus, and Adams consensus are not [6].…”