  We revisit the discrete additive and multiplicative coalescents, starting
with $n$ particles with unit mass. These cases are known to be related to some
"combinatorial coalescent processes": a time reversal of a fragmentation of
Cayley trees or a parking scheme in the additive case, and the random graph
process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding
these combinatorial objects in real-valued processes indexed by the line is the
key to describing the asymptotic behaviour of the masses as $n\to +\infty$.
  We propose to use the Prim order on the vertices instead of the classical
breadth-first (or depth-first) traversal to encode the combinatorial coalescent
processes. In the additive case, this yields interesting connections between
the different representations of the process. In the multiplicative case, it
allows one to answer to a stronger version of an open question of Aldous [Ann.
Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of
(rescaled) masses, seen as a process indexed by the time $\lambda$, converges
in distribution to the reordered sequence of lengths of the excursions above
the current minimum of a Brownian motion with parabolic drift $(B_t+\lambda t -
t^2/2, t\geq 0)$, but we also construct a version of the standard augmented
multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel.,
to appear] using an additional Poisson point process.

Probability;Discrete Mathematics;Combinatorics