Given a triangulation (without Steiner points) of a simple polygon $P$, one can consider the dual of this triangulation, which is defined as follows. We create a vertex for every triangle in our triangulation, and we connect two vertices if the corresponding triangles share an edge. The dual graph is known to be a tree with maximum degree three.For my application, I am interested in the following. Given a tree $T$ with maximum degree three, is there always a simple polygon $P$ such that the dual of every triangulation (without Steiner points) of $P$ is equal to $T$. Here, the triangulation of $P$ may not be unique, but I require that the dual graph be unique.This is certainly true when $T$ is a path, but becomes unclear when you have vertices of degree three.
Unique triangulation duals of simple polygons
reference request;computational geometry