A polytope plays a central role in different areas of mathematics. It is used quite heavily in applied fields of mathematics, such as medical imaging and robotics, geometric modeling.
A polytope has many users in modern science such as computer graphics, optimization, and search engine. It is intended for a broad audience of mathematically inclined. Therefore in this paper, we shall take a polytope with one kind of application, which is known as a Stasheff polytope. it has been applied in: moduli spaces, Erhart polynomial, physics- chemistry, and Hopf algebras.
In this paper,a finite graph G with tubingis taken, the nodes of the graph are reducetopoints in R^n and the convex hull for them are simplex, permutahedron and associahedron (Stasheff polytope ) are studied.
Also, a fan graph to the painted tree is also taken and reduces its nodes to points inR^n. The converse of this result is also given with different examples to consult our results.