  Symmetric Dyck tilings and ballot tilings are certain tilings in the region
surrounded by two ballot paths. We study the relations of combinatorial objects
which are bijective to symmetric Dyck tilings such as labeled trees, Hermite
histories, and perfect matchings. We also introduce two operations on labeled
trees for symmetric Dyck tilings: symmetric Dyck tiling strip (symDTS) and
symmetric Dyck tiling ribbon (symDTR). We give two definitions of Hermite
histories for symmetric Dyck tilings, and show that they are equivalent by use
of the correspondence between symDTS operation and an Hermite history. Since
ballot tilings form a subset in the set of symmetric Dyck tilings, we construct
an inclusive map from labeled trees for ballot tilings to labeled trees for
symmetric Dyck tilings. By this inclusive map, the results for symmetric Dyck
tilings can be applied to those of ballot tilings. We introduce and study the
notions of ballot tableaux and tree-like tableaux of shifted shapes, which are
generalizations of Dyck tableaux and tree-like tableaux, respectively. The
correspondence between ballot tableaux and tree-like tableaux of shifted shapes
is given by using the symDTR operation and the structure of labeled trees for
symmetric Dyck tilings.
