Authors: Margaret Archibald ¹; Julien Clément ^2,3

• 1 School of Mathematics [Johannesburg]
• 2 Equipe AMACC - Laboratoire GREYC - UMR6072
• 3 Laboratoire d'Informatique Gaspard-Monge

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.