On the Average Profile of Symmetric Digital Search Trees∗

A digital search tree (DST) – one of the most fundamental data structures on words – is a digital tree in which keys (strings, words) are stored directly in (internal) nodes. The profile of a digital search tree is a parameter that counts the number of nodes at the same distance from the root. It is a function of the number of nodes and the distance from the root. Several tree parameters, such as height, size, depth, shortest path, and fill-up level, can be uniformly analyzed through the profile. In this note we analyze asymptotically the average profile for a symmetric digital search tree in which strings are generated by an unbiased memoryless source. We show that the average profile undergoes several phase transitions: initially it resembles a full tree until it starts growing polynomially with the number of nodes, and then it decays first polynomially, then exponentially, and finally quadratic exponentially. We derive these results by a combinational of analytic techniques such as saddle point method and ideas of applied mathematics such as linearization.