On 2-protected nodes in random digital trees

In this paper, we consider the number of 2-protected nodes in random digital trees. For tries, Gaither, Homma, Sellke and Ward (2012) and Gaither and Ward (2013) derived asymptotic expansions for mean and variance. We re-derive their results (with some corrections) and show their conjectured central limit theorem. This will be done by recent unified approaches by Fuchs, Hwang and Zacharovas (2014) and Fuchs and Lee (2014). Interestingly, the expressions we obtain are quite different from previous ones and contain divergent series which have values by appealing to the theory of Abel summability. Similar results will be shown for the number of 2-protected notes in PATRICIA tries, too. Moreover, we will also add asymptotic expansions for the variance and a central limit theorem to the known expansion for the mean of 2-protected nodes in digital search trees due to Du and Prodinger (2012).