  We study the maximum numbers of pseudo-triangulations and pointed
pseudo-triangulations that can be embedded over a specific set of points in the
plane or contained in a specific triangulation.
  We derive the bounds $O(5.45^N)$ and $\Omega (2.41^N)$ for the maximum number
of pointed pseudo-triangulations that can be contained in a specific
triangulation over a set of $N$ points. For the number of all
pseudo-triangulations contained in a triangulation we derive the bounds
$O^*(6.54^N)$ and $\Omega (3.30^N)$. We also prove that $O^*(89.1^N)$ pointed
pseudo-triangulations can be embedded over any specific set of $N$ points in
the plane, and at most $120^N$ general pseudo-triangulations.
