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We introduce an object called a tree growing sequence (TGS) in an effort togeneralize bijective correspondences between $G$-parking functions, spanningtrees, and the set of monomials in the Tutte polynomial of a graph $G$. A treegrowing sequence determines an algorithm which can be applied to a singlefunction, or to the set $\mathcal{P}_{G,q}$ of $G$-parking functions. When thelatter is chosen, the algorithm uses splitting operations - inspired by therecursive defintion of the Tutte polynomial - to iteratively break$\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective maps$\tau$ and $\rho$ from $\mathcal{P}_{G,q}$ to the spanning trees of $G$ andTutte monomials, respectively. We compare the TGS algorithm to Dhar's algorithmand the family described by Chebikin and Pylyavskyy. Finally, we compute aTutte polynomial of a zonotopal tiling using analogous splitting operations.

Generalized Bijective Maps between $G$-Parking Functions, Spanning Trees, and the Tutte Polynomial