Distances in random digital search trees

Distances between nodes in random trees is a popular topic, and several classes of trees have recently been investigated. We look into this matter in digital search trees. By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. In addition to various asymptotics, we give an exact expression for the mean and for the variance in the unbiased case. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; it is prudent to seek a shortcut to the limit via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution of a distributional equation (distances being measured in the Wasserstein metric space). An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.