  Motivated by the question of how macromolecules assemble, the notion of an
{\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is
concerned with enumerating the number of assembly trees of $G$, a problem that
applies to the macromolecular assembly problem. Explicit formulas or generating
functions are provided for the number of assembly trees of several families of
graphs, in particular for what we call $(H,\phi)$-graphs. In some natural
special cases, we apply powerful recent results of Zeilberger and Apagodu on
multivariate generating functions, and results of Wimp and Zeilberger, to
deduce recurrence relations and very precise asymptotic formulas for the number
of assembly trees of the complete bipartite graphs $K_{n,n}$ and the complete
tripartite graphs $K_{n,n,n}$. Future directions for reseach, as well as open
questions, are suggested.
