Dyck tilings of type $D$

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.