  We explore a generating function trick which allows us to keep track of
infinitely many statistics using finitely many variables, by recording their
individual distributions rather than their joint distributions. Building on
previous work of Panholzer and Prodinger, we apply this method to study the
depth distributions of individual nodes in rooted binary trees, plane trees,
noncrossing trees, and increasing trees; the height distributions of individual
vertices and individual steps in Dyck paths; and the number of diagonals
separating two fixed sides of a convex polygon in a triangulation or
dissection. We obtain both exact and asymptotic results, which sometimes refine
known formulas or provide combinatorial proofs of results from the probability
literature.
