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.