Alarm bells for the molecular clock? No support for Ho et al.'s model of time-dependent molecular rate estimates.

trees are binary, split fit is co-Pareto on components (Wilkinson et al., 2005a). When input trees have polytomies, an optimal split fit supertree could include arbitrary resolutions that do not contradict any displayed splits. In that case, split fit can fail to be co-Pareto (but not sub-Pareto) on components. In contrast, the corresponding strict consensus, the split fit* supertree, suppresses all arbitrary resolutions and must be co-Pareto on components. A method that is co-Pareto on components must also be co-Pareto (and sub-Pareto) on triplets and on nestings. Our example confirms that triplet fit is not co-Pareto on components. For example, of the 1677 triplet MRP supertrees, 339 include the component (DE). This is not present in either input tree but all the triplets that it entails are present in at least one of the input trees. It also confirms that that the MinCut methods are not co-Pareto on components (Semple and Steel, 2000). For example, D and K are grouped together in the MinCut supertree (Fig. lc) and D and K nest within A-P in both input trees but they are not a component in either input tree (Fig. 1). Co-Pareto on triplets.—The triplet fit method is analogous to split fit, and by analogy we might expect that triplet fit* is co-Pareto and thus sub-Pareto on triplets, and that triplet fit is sub-Pareto on triplets and co-Pareto on triplets when input trees are binary (in which case there are no arbitrary resolutions). However, unlike full splits, combinations of displayed triplets can entail triplets that are not displayed. Thus, the analogy breaks down and there are, as yet, no proofs for any of these conjectures, which remain open problems. Similarly, quartet fit' is conjectured but has not been shown to be co-Pareto and sub-Pareto on quartets.