  We generalize the concept of ascending and descending runs from permutations
to rooted labelled trees and mappings, i.e., functions from the set $\{1,
\dots, n\}$ into itself. A combinatorial decomposition of the corresponding
functional digraph together with a generating functions approach allows us to
perform a joint study of ascending and descending runs in labelled trees and
mappings, respectively. From the given characterization of the respective
generating functions we can deduce bivariate central limit theorems for these
quantities. Furthermore, for ascending runs (or descending runs) we gain
explicit enumeration formulae showing a connection to Stirling numbers of the
second kind. We also give a bijective proof establishing this relation, and
further state a bijection between mappings and labelled trees connecting the
quantities in both structures.
