On protected nodes in Digital Search Trees

Recently, 2-protected nodes were studied in the context of ordered trees and $k$-trees. These nodes have a distance of at least 2 to each leaf. Here, we study digital search trees, which are binary trees, but with a different probability distribution underlying. Our result says, that \emph{grosso modo} some 31% of the nodes are 2-protected. Methods include exponential generating functions, contour integration, and some elements from $q$-analysis.