Patchy taxon coverage, and lateral gene transfer are two phenomena that can obscure signal for an underlying species tree. In the first part of this talk, I describe the extent to which pattern of taxon coverage can be ‘phylogenetically decisive’, and provide some applications to data.  In part 2, I outline recent work on the problem of inferring a ‘species tree’ from gene trees, under a simple model of random lateral gene transfer (LGT).  Two questions here are ‘does the most probable gene tree topology always agree with the species tree?’ and ‘how much transfer is possible while still retaining signal for a central tendency tree?’  Our analysis involves a curious connection to random walks on cyclic graphs.