The Variance of Level Numbers in Certain Families of Trees

For the families of t-ary trees and planted plane trees, the variance of the level numbers (i.e., the depths of specified endnodes) is determined. We apply different types of techniques, such as singularity analysis (Kirschenhofer's diagonalization method) or a probabilistic approach using weak convergence and uniform integrability. In the case of binary trees, an exact variance formula for finite tree size is established; in the other cases, asymptotic equivalents are derived. Also some results on higher moments are presented.