Operads and phylogenetic trees

We construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of Com, the operad for commutative semigroups, and [0,∞), the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of n-ary operations of Phyl and Tn × [0,∞)n+1, where Tn is the space of metric n-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of Phyl. These always extend to coalgebras of the larger operad Com + [0,∞], since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad O, its coproduct with [0,∞] contains the operad W (O) constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees.