  Asymptotics of the variances of many cost measures in random digital search
trees are often notoriously messy and involved to obtain. A new approach is
proposed to facilitate such an analysis for several shape parameters on random
symmetric digital search trees. Our approach starts from a more careful
normalization at the level of Poisson generating functions, which then provides
an asymptotically equivalent approximation to the variance in question. Several
new ingredients are also introduced such as a combined use of the Laplace and
Mellin transforms and a simple, mechanical technique for justifying the
analytic de-Poissonization procedures involved. The methodology we develop can
be easily adapted to many other problems with an underlying binomial
distribution. In particular, the less expected and somewhat surprising $n(\log
n)^2$-variance for certain notions of total path-length is also clarified.
