  In 1960, R\'enyi asked for the number of random queries necessary to recover
a hidden bijective labeling of n distinct objects. In each query one selects a
random subset of labels and asks, what is the set of objects that have these
labels? We consider here an asymmetric version of the problem in which in every
query an object is chosen with probability p > 1/2 and we ignore "inconclusive"
queries. We study the number of queries needed to recover the labeling in its
entirety (the height), to recover at least one single element (the fillup
level), and to recover a randomly chosen element (the typical depth). This
problem exhibits several remarkable behaviors: the depth D_n converges in
probability but not almost surely and while it satisfies the central limit
theorem its local limit theorem doesn't hold; the height H_n and the fillup
level F_n exhibit phase transitions with respect to p in the second term. To
obtain these results, we take a unified approach via the analysis of the
external profile, defined at level k as the number of elements recovered by the
kth query. We first establish new precise asymptotic results for the average
and variance, and a central limit law, for the external profile in the regime
where it grows polynomially with n. We then extend the external profile results
to the boundaries of the central region, leading to the solution of our problem
for the height and fillup level. As a bonus, our analysis implies novel results
for analogous parameters of random PATRICIA tries.
