Node Profiles of Symmetric Digital Search Trees

We give a detailed analysis in distribution of the profiles of random symmetric digital search trees, which are in close connection with the performance of the search complexity of random queries in such trees. While the expected profiles have been analyzed for several decades, the analysis of the variance turns out to be very difficult and challenging, and requires the combination of many new techniques. We also establish by contraction method the asymptotic normality of the profiles in the range where the variance tends to infinity. As consequences of our results, we obtain a two-point concentration for the distributions of the height and the saturation level. AMS 2010 subject classifications. Primary 05A16,60C05, 68Q25; secondary 68P05, 60F05.