Triangulations with few ears: symmetry classes and disjointness

An ear in a triangulation $T$ of a convex $n$-gon $P$ is a triangle of $T$ that shares two sides with $P$ itself. Certain enumerational and structural problems become easier when one considers only triangulations with few ears. We demonstrate this in two ways. First, for $k=2, 3$, we find the number of symmetry classes of triangulations with $k$ ears. Second, for $k=2, 3$, we determine the number of triangulations disjoint from a given triangulation: this number depends only on $n$ for $k=2$, and only on lengths of branches of the dual tree for $k=3$.