  We consider a class of density-dependent branching processes which
generalises exponential, logistic and Gompertz growth. A population begins with
a single individual, grows exponentially initially, and then growth may slow
down as the population size moves towards a carrying capacity. A fixed number
of individuals are sampled at a time while the population is still growing
superlinearly. Taking the sampling time and carrying capacity simultaneously to
infinity, we prove convergence of the coalescent tree. The limiting coalescent
tree is in a sense universal over our class of models.
