Although I am not sure I fully understand the question, perhaps this attempt at a partial
answer will clarify matters.
First, it is known that every tree of at least one edge and with
no vertices of degree $2$ can be realized as the Voronoi diagram of a
set of points in convex position. See Ref.(1) below, from which this figure is taken:
Second, the general question of recognizing when a convex partition of the plane
is a Voronoi diagram of some set of points was explored all the way back in 1985,
Ref.(2) below, which solved it for the special case when all Voronoi vertices have degree 3. For the general, unrestricted case, recognition can be achieved in polynomial time via linear programming, Ref.(3).
But perhaps reducing the question to linear programming does not constitute an "interesting" characterization. I think the Ash-Bolker paper might include
an interesting characterization, but I can't quite remember their argument...
(1) G. Liotta and H. Meijer, “Voronoi Drawings of Trees,” Comput. Geom.
Theory and Appl., vol. 24, no. 3, pp. 147–178, 2003. (Journal link)
(2) P. Ash and E. Bolker, “Recognizing Dirichlet Tesselations,” Geometriae
Dedicata, vol. 19, pp. 175–206, 1985.
(3) D. Hartvigsen, “Recognizing Voronoi Diagrams with Linear Programming,” ORSA J. Computing, vol. 4, no. 4, pp. 369–374, 1992. (Journal link)