Trees are not just combinatorial structures: they are also
biological structures, both in the obvious way but also in the
study of evolution. Starting from DNA samples from living
species, biologists use increasingly sophisticated mathematical
techniques to reconstruct the most likely “phylogenetic tree”
describing how these species evolved from earlier ones. In their
work on this subject, they have encountered an interesting
example of an operad, which is obtained by applying a variant of
the Boardmann–Vogt “W construction” to the operad for
commutative monoids. The operations in this operad are labelled
trees of a certain sort, and it plays a universal role in the
study of stochastic processes that involve branching. It also
shows up in tropical algebra. This talk is based on work in
progress with Nina Otter [www.fair-fish.ch].