  We study combinatorial properties of a rational Dyck path by decomposing it
into a tuple of Dyck paths. The combinatorial models such as $b$-Stirling
permutations, $(b+1)$-ary trees, parenthesis presentations, and binary trees
play central roles to establish a correspondence between the rational Dyck path
and the tuple of Dyck paths. We reinterpret two orders, the Young and the
rotation orders, on rational Dyck paths in terms of the tuple of Dyck paths by
use of the decomposition. As an application, we show a duality between
$(a,b)$-Dyck paths and $(b,a)$-Dyck paths in terms of binary trees.
