Phylogenomic data often exhibit partial taxon coverage, whereby each loci is present or sequenced only for some corresponding subset of the species under study. This leads to some interesting mathematical and statistical questions as to whether a fully resolved underlying evolutionary tree for all the taxa can be reconstructed, given perfect phylogenetic estimates from each locus. We first describe the extent to which a pattern of taxon coverage can be 'phylogenetically decisive' in various senses, and provide some applications to data (joint work with Sanderson and McMahon). We then consider tree reconstruction from distance data, in settings where accurate estimates of evolutionary distance exist between only certain pairs of taxa: Does this partial information suffice to pin down - or, as we say, 'lasso' - the underlying phylogenetic tree? We describe a number of new combinatorial results concerning lassos (joint work with Dress and Huber). We conclude with some open problems.
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