Analysis of some parameters for random nodes in priority trees

Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers $1, dots, n$ are equally likely to construct a priority tree of size $n$ we study the following parameters in size-$n$ trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.