The associahedron is the -dimensional generalization of the pentagon. It was discovered by Stasheff in 1963 and it is also known as the Stasheff polytope. The number of nodes in the -associahedron is equivalent to the number of binary trees with nodes, which is the Catalan number .
The associahedron is the basic tool in the study of homotopy associative Hopf spaces.
Loday (2004) provides the following method for associahedron construction. Take , the set of planar binary trees with leaves. Define as the number of leaves to the left of the th vertex and as the number of leaves to the right of the th vertex. For in , define
The -associahedron is then defined as the convex hull of .
The associahedron can be obtained by removing facets from the permutohedron, and is related to the cyclohedron and permutohedron.