  We introduce a correspondence between phylogenetic trees and Brauer diagrams,
inspired by links between binary trees and matchings described by Diaconis and
Holmes (1998). This correspondence gives rise to a range of semigroup
structures on the set of phylogenetic trees, and opens the prospect of many
applications. We furthermore extend the Diaconis-Holmes correspondence from
binary trees to non-binary trees and to forests, showing for instance that the
set of all forests is in bijection with the set of partitions of finite sets.
