  We prove that the Barab\'asi-Albert model converges weakly to a set of
generalized Yule models via an appropriate scaling. To pursue this aim we
superimpose to its graph structure a suitable set of processes that we call the
planted model and we introduce an ad-hoc sampling procedure. The use of the
obtained limit process represents an alternative and advantageous way of
looking at some of the asymptotic properties of the Barab\'asi-Albert random
graph.
