Simple models of speciation/extinction make various predictions about the shape of a phylogenetic tree (such trees represent the evolutionary history of a group of extant species). Conversely, by looking at the shape of reconstructed phylogenetic trees one might hope to test various hypotheses concerning an underlying speciation process, particularly
when the number of extant species is large.  In this talk we present a brief overview of this area, along with some recent and new results.  We outline some connections to processes and techniques that have also been useful in other settings (including statistical physics) - in particular (extended) Polya-Urn models, coalescent processes, and generating functions - and show how these links can provide exact and asymptotic solutions to some of the questions posed.

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