On 2-protected nodes in random digital trees

In this paper, we consider the number of 2-protected nodes in random digital trees. Results for the mean and variance of this number for tries have been obtained by Gaither et al. (2012) [11] and Gaither and Ward (2013) [10] and for the mean in digital search trees by Du and Prodinger (2012) [5]. In this short note, we show that these previous results and extensions such as the variance in digital search trees and limit laws in both cases can be derived in a systematic way by recent approaches of Fuchs et al. (2012; 2014) [8], [15] and Fuchs and Lee (2014) [9]. Interestingly, the results for the moments we obtain by our approach are quite different from the previous ones and contain divergent series which have values by appealing to the theory of Abel summability. We also show that our tools apply to PATRICIA tries, for which the number of 2-protected nodes has not been investigated so far.