  We investigate vertex levels of containment in a random hypergraph grown in
the spirit of a recursive tree. We consider a local profile tracking the
evolution of the containment of a particular vertex over time, and a global
profile concerned about counts of the number of vertices of a particular
containment level.
  For the local containment profile, we obtain the exact mean, variance and
probability distribution in terms of standard combinatorial quantities like
generalized harmonic numbers and Stirling numbers of the first kind.
Asymptotically, we observe phases: the early vertices have an asymptotically
normal distribution, intermediate vertices have a Poisson distribution, and
late vertices have a degenerate distribution.
  As for the global containment profile, we establish an asymptotically normal
distribution for the number of vertices at the smallest containment level as
well as their covariances with the number of vertices at the second smallest
containment level and the variances of these numbers.
