  This paper uses combinatorics and group theory to answer questions about the
assembly of icosahedral viral shells. Although the geometric structure of the
capsid (shell) is fairly well understood in terms of its constituent subunits,
the assembly process is not. For the purpose of this paper, the capsid is
modeled by a polyhedron whose facets represent the monomers. The assembly
process is modeled by a rooted tree, the leaves representing the facets of the
polyhedron, the root representing the assembled polyhedron, and the internal
vertices representing intermediate stages of assembly (subsets of facets).
Besides its virological motivation, the enumeration of orbits of trees under
the action of a finite group is of independent mathematical interest. If $G$ is
a finite group acting on a finite set $X$, then there is a natural induced
action of $G$ on the set $\mathcal{T}_X$ of trees whose leaves are bijectively
labeled by the elements of $X$. If $G$ acts simply on $X$, then $|X| := |X_n| =
n \cdot |G|$, where $n$ is the number of $G$-orbits in $X$. The basic
combinatorial results in this paper are (1) a formula for the number of orbits
of each size in the action of $G$ on $\mathcal{T}_{X_n}$, for every $n$, and
(2) a simple algorithm to find the stabilizer of a tree $\tau \in
\mathcal{T}_X$ in $G$ that runs in linear time and does not need memory in
addition to its input tree.
