  We investigate the distribution of the depth of a node containing a specific
key or, equivalently, the number of steps needed to retrieve an item stored in
a randomly grown binary search tree. Using a representation in terms of mixed
and compounded standard distributions, we derive approximations by Poisson and
mixed Poisson distributions; these lead to asymptotic normality results. We are
particularly interested in the influence of the key value on the distribution
of the node depth. Methodologically our message is that the explicit
representation may provide additional insight if compared to the standard
approach that is based on the recursive structure of the trees. Further, in
order to exhibit the influence of the key on the distributional asymptotics, a
suitable choice of distance of probability distributions is important. Our
results are also applicable in connection with the number of recursions needed
in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm Find.
