Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from $0$ to $n$. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with $n+2$ vertices. The asymptotic behaviour of this distribution is described in a limiting regime where the distribution of the delabeled fringe subtree approaches that of a Galton-Watson tree with a mixed Poisson offspring distribution.