on the species tree, suggesting that the standard majority-rule consensus (which only accepts clades occurring more than 50% of the time) is very conservative in this setting. If the greedy consensus approach is used for clades with probability greater than 1/3 (leaving the tree unresolved with respect to clades with lower probability), then this “not-too-greedy" consensus approach is not misleading, in the sense that it asymptotically cannot return a false species tree clade as the number of loci approaches infinity.

In contrast, previous results have shown that when greedy consensus is applied without restrictions on clade probabilities, the returned tree can be misleading (*i.e.*, for some species trees, as the number of loci increases, the greedy consensus method is increasingly likely to produce a tree that disagrees with the true species tree) for some sets of branch lengths [@degnanEtAl2009]. These “too-greedy zones" of edge lengths occur on $4$-taxon asymmetric species trees and on any species tree topology with five or more leaves. Thus, caution must be used when probabilities of clades are less than 1/3; it is not obvious how to determine which low-probability clades are on the species tree, even if clade probabilities are known exactly. Other examples show 