Orthogeodesic point-set embedding of trees

We study the following problem. Given a family F of trees, what is the minimum value f(n) such that every n-vertex tree in F admits an orthogeodesic point-set embedding on every set of grid points of size f(n) such that no two points lie on the same horizontal or vertical line? We provide polynomial upper bounds on f(n) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped.