Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.

N. BROUTIN;L. DEVROYE, 2008, An Analysis of the Height of Tries with Random Weights on the Edges, Combinatorics Probability Computing, 17, 2, pp. 161-202, 10.1017/s0963548307008796.