  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. At a time while
the population is still growing superlinearly, a fixed number of individuals
are sampled and their coalescent tree is drawn. Taking the sampling time and
carrying capacity simultaneously to infinity, we prove convergence of the
coalescent tree to a limiting tree which is in a sense universal over our class
of models.
