Properties of supertree methods in the consensus setting.

Supertree methods (SMs) are techniques for inferring (super)trees from sets of (input) trees. Classical consensus methods are SMs that were designed for the special case where input trees have identical leaf sets. The need for methods that can also combine information from input trees with nonidentical leaf sets has led to many alternative SMs. Some of these SMs are generalizations from conservative consensus methods (strict and semistrict) that do not resolve input tree conflicts (e.g., Gordon, 1986; Goloboff and Pol, 2002). Our focus here is on more liberal SMs, those capable of resolving conflicts among input trees. Liberal SMs comprise the majority of described methods and have been the most used in practice by biologists seeking well-resolved phylogenies. However, today's practitioners are confronted with choosing among a potentially bewildering array of liberal SM(s). Wilkinson et al. (2004) argued that nonarbitrary, rational choice among liberal SMs would best be guided by knowledge of the comparative accuracy of alternative methods. However, there have been few comparisons of accuracy using simulations (Bininda-Emonds and Sanderson, 2001; Chen et al., 2003; Eulenstein et al., 2004; Lapointe and Levasseur, 2004; Ross and Rodrigo, 2004) over a restricted range of conditions. Thus, Wilkinson et al. (2004) also discussed a number of properties that they suggested provide surrogates for accuracy and might therefore be expected of any SM. One of these, that input tree conflicts should be resolved independently of input tree shape, was investigated by Wilkinson et al. (2005a), who used a simple example (Fig. 1) and simulations to demonstrate input tree shape effects with 8 of the 14 methods they investigated, including the widely used matrix representation with parsimony (MRP) of Baum (1992) and Ragan (1992). Here we introduce a class of sub-Pareto properties that we argue constitute particularly weak expectations of how accurate SMs should handle consensus problems. We then use the same example to substantiate and extend results reported in Wilkinson et al. (2004) and to demonstrate that seven of the liberal SMs that are sensitive to input tree shape also lack some seemingly reasonable consensus properties. Lastly, we consider the relevance of these properties to choice and PRELIMINARIES