Spanning trees in random series-parallel graphs

By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on n vertices chosen uniformly at random satisfies an estimate of the formsϱ−n(1+o(1)),sϱ−n(1+o(1)), where s and ϱ   are computable constants, the values of which are approximately s≈0.09063s≈0.09063 and ϱ−1≈2.08415ϱ−1≈2.08415. We obtain analogous results for subfamilies of series-parallel graphs including 2-connected series-parallel graphs, 2-trees, and series-parallel graphs with fixed excess.