Abstract
We construct a family of extremely simple bijections that yield Cayley's famous formula for counting trees. The weight preserving properties of these bijections furnish a number of multivariate generating functions for weighted Cayley trees. Essentially the same idea is used to derive bijective proofs and q -analogues for the number of spanning trees of other graphs, including the complete bipartite and complete tripartite graphs. These bijections also allow the calculation of explicit formulas for the expected number of various statistics on Cayley trees.
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