  The basis of this paper is the elementary observation that the $n$-step
descendant distribution of any Galton-Watson process satisfies a discrete
Smoluchowski coagulation equation with multiple coalescence. Using this we
obtain necessary and sufficient criteria for the convergence of scaling limits
of Galton-Watson processes that are simpler (and equivalent) to the classical
criteria obtained by Grimvall in 1974. Our results provide a clear and natural
interpretation, and an alternate proof, of the fact that the L\'evy jump
measure of certain CSBPs satisfies a generalized Smoluchowski equation. (This
result was previously proved by Bertoin and Le Gall in 2006.)
  Moreover, our analysis shows that the nonlinear scaling dynamics of CSBPs
becomes linear and purely dilatational when expressed in terms of the L\'evy
triple associated with the branching mechanism. We use this to prove existence
of universal critical Galton-Watson and CSBPs analogous to W. Doeblin's
"universal laws". Namely, these universal processes generate all possible
critical and subcritical CSBPs as subsequential scaling limits.
  Our convergence results rely on a natural topology for L\'evy triples and a
continuity theorem for Bernstein transforms (Laplace exponents). We develop
these in a self-contained appendix.
