  Sandpile groups are a subtle graph isomorphism invariant, in the form of a
finite abelian group, whose cardinality is the number of spanning trees in the
graph. We study their group structure for graphs obtained by attaching a cone
vertex to a tree. For example, it is shown that the number of generators of the
sandpile group is at most one less than the number of leaves in the tree. For
trees on a fixed number of vertices, the paths and stars are shown to provide
extreme behavior, not only for the number of generators, but also for the
number of spanning trees, and for Tutte polynomial evaluations that count the
recurrent sandpile configurations by their numbers of chips.
