  We introduce and study cover-inclusive and cover-exclusive Dyck tilings of
type $D$. It is shown that the generating functions of Dyck tilings of type $D$
are expressed in terms of the generating function of ballot tilings of type
$B$. We introduce link patterns of type $D$ and plane trees for a ballot path,
and construct a map from trees to $\mathbb{Z}[q]$. This map gives the
generating function of cover-inclusive Dyck tilings of type $D$ associated to
the ballot path.
