Simple but Fundamental Limitations on Supertree and Consensus Tree Methods

Abstract

363 Ragan (1992), Purvis (1995a), and Sanderson et al. (1998). The purpose of this note is to point out that inherent limitations exist as to what any supertree or consensus tree method can possibly achieve. More precisely, we demonstrate here by simple examples that, although one can easily list various desirable properties that one would like such a method to possess, no method can possess all of these simultaneously. Although this may be viewed as a somewhat negative result, its positive spin-off is that it focuses attention on those types of supertree constructions one should aim for and indicates promising lines for future work. Although we insist that any supertree method must return a single tree, we will briefly discuss extensions that allow several trees to be in the output. One inherent limitation on the solution to the consensus tree problem for unrooted trees has been described by McMorris (1985), who discussed three conditions that cannot simultaneously be satisfied by any consensus method. The conditions we consider here are quite different and somewhat simpler. We will also see that there is a fundamental distinction between rooted and unrooted trees.