  We provide a new succession rule (i.e. generating tree) associated with
Schr\"oder numbers, that interpolates between the known succession rules for
Catalan and Baxter numbers. We define Schr\"oder and Baxter generalizations of
parallelogram polyominoes, called slicings, which grow according to these
succession rules. In passing, we also exhibit Schr\"oder subclasses of Baxter
classes, namely a Schr\"oder subset of triples of non-intersecting lattice
paths, a new Schr\"oder subset of Baxter permutations, and a new Schr\"oder
subset of mosaic floorplans. Finally, we define two families of subclasses of
Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings,
for $m \in \mathbb{N}$. Using functional equations and the kernel method, their
generating functions are computed in some special cases, and we conjecture that
they are algebraic for any $m$.
