  We give an asymptotic expression for the expected number of spanning trees in
a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$,
provided that the number of edges is at least $n + \textstyle{\frac{1}{2}}
d_{\max}^4$, where $d_{\max}$ is the maximum degree. A key part of our argument
involves establishing a concentration result for a certain family of functions
over random trees with given degrees, using Pr\"ufer codes.
