for a given elliptic curve $e$ over a finite field $f_q$, we let $n_k = \#e(f_{q^k})$, where $f_{q^k}$ is a $k$th degree extension of the finite field $f_q$. because the zeta function for $e$ only depends on $q$ and $n_1$, the sequence $\{n_k\}$ only depends on those numbers as well. more specifically, we observe that these bivariate expressions for $n_k$ are in fact polynomials with integer coefficients, which alternate in sign with respect to the power of $n_1$. this motivated a search for a combinatorial interpretation of these coefficients, and one such interpretation involves spanning trees of a certain family of graphs. in this talk, i will describe this combinatorial interpretation, as well as applications and directions for future research. this will include determinantal formulas for $n_k$, factorizations of $n_k$, and the definition of a new sequence of polynomials, which we call elliptic cyclotomic polynomials. one of the important features of elliptic curves which makes them the focus of contemporary research is that they admit a group structure. during the remainder of this talk i will describe chip-firing games, how they provide a group structure on the set of spanning trees, and numerous ways that these groups are analogous to those of elliptic curves.