  We present a detailed study of how dark matter haloes assemble their mass and
grow their (central) potential well. We characterize these via their mass
accretion histories (MAHs) and potential well growth histories (PWGHs), which
we extract from the Bolshoi simulation and from semi-analytical merger trees
supplemented with a method to compute the maximum circular velocity, Vmax, of
progenitor haloes. The results of both methods are in excellent agreement, both
in terms of the average and the scatter. We show that the MAH and PWGH are
tightly correlated, and that growth of the central potential precedes the
assembly of mass; the maximum circular velocity is already half the present day
value by the time the halo has accreted only 2 percent of its final mass.
Finally, we demonstrate that MAHs have a universal form, which we use to
develop a new and improved universal model that can be used to compute the
average or median MAH and PWGH for a halo of any mass in any LCDM cosmology,
without having to run a numerical simulation or a set of halo merger trees.
