I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra. I have some questions, that might have an answer, just I am not aware of it.

Assume, that we are working in 3dimensional Euclidean space, with equilateral simplices. I would like to glue together tetrahedra in a combinatorial way, so do not allow for degenerate triangulations. I define the area (A) as the triangles on the boundary of the triangulation. The geometry that maximises the boundary will be a tree of tetrahedra, so the maximal edge degree (number of tetrahedra around an edge) will be 3.

Is it known how many triangulations are there inside a triangulated sphere with boundary $A$?