Persi Diaconis and Susan Holmes showed in 1998 that binary phylogenetic trees on $n$ leaves can be encoded by matchings on a set of $2n-2$ elements, that is, a partition of the set into pairs.  In this talk I will describe how this can be extended to correspondences for non-binary phylogenetic trees and forests, and for phylogenetic networks.  For instance, the set of phylogenetic forests is in bijection with the set of all partitions of finite sets, and a large class of networks (the ``labellable'' networks) is in bijection with the set of ``expanding'' covers of finite sets.  In some cases, these set-theoretic correspondences align with Brauer and partition diagrams, which have a monoid structure, and can even give rise to a product on tree-space via the ``sandwich semigroup'', posing some interesting possible lines of inquiry.  Joint work with Peter Jarvis (Tasmania), Mike Steel (Canterbury), and Daniele Marchei (Camerino).

Afternoon tea will be provided at 3:30pm