Dated species tree and biogeographic inferences for Hydatellaceae. (a) Timing of divergences in Hydatellaceae inferred using a multi-species coalescent analysis, based on four plastid genes and nuclear ITS regions, with prior dating estimates for the first two nodes derived from a separate seed-plant analysis (see Additional file 3; labelled nodes are referred to in Additional file 5). Numbers beside branches are support values (posterior probabilities expressed as percentages); dashes indicate <50% support. Divergence time uncertainty is noted by blue bars, representing 95% HPD. The time scale is in Ma. Letters adjacent to tips represent: E = east, T. =Trithuria, W = west. Historical biogeography inferred in (b) with the full model of ML ancestral-state reconstruction, in (c) with MP based dispersal-vicariance analysis, and in (d) with ML based dispersal-extinction cladogenesis analysis. Pie fractions in (b) represent relative likelihoods; in (c) and (d) they represent joint areas where the species is inferred to have existed in multiple areas. The relative likelihood of the best geographic range pair is shown in (d) adjacent to individual nodes.