A family of bijections between -parking functions and spanning trees

For a directed graph G on vertices { 0 , 1 , … , n } , a G-parking function is an n-tuple ( b 1 , … , b n ) of non-negative integers such that, for every non-empty subset U ⊆ { 1 , … , n } , there exists a vertex j ∈ U for which there are more than b j edges going from j to G - U . We construct a family of bijective maps between the set P G of G-parking functions and the set T G of spanning trees of G rooted at 0, thus providing a combinatorial proof of | P G | = | T G | .