We consider a family of growth models defined using conformal maps in which the local growth rate is determined by |phi n '|-eta where phi n is the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for eta>1aggregating particles attach to their immediate predecessors with high probability, while for eta<1almost surely this does not happen.