  We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled
trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$,
the operad for commutative semigroups, and $[0,\infty)$, 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 $\mathrm{Phyl}$ and $\mathcal{T}_n\times [0,\infty)^{n+1}$, where
$\mathcal{T}_n$ 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 $\mathrm{Phyl}$. These
always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$,
since Markov processes on finite sets converge to an equilibrium as time
approaches infinity. We show that for any operad $O$, its coproduct with
$[0,\infty]$ 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.
