I cannot find where I learned that the $n$-dimensional associahedron is a union of $n$-cubes. The vertices of the $n$-dimensional associahedron are the finite binary trees having $n+2$ leaves and there is one cube for each vertex (tree). If the internal edges of the tree (those edges whose endpoints both do not have valence one) are allowed to independently take on any length in the unit interval $[0,1]$, then these "distorted" trees parametrize the cube. (There are $n$ internal edges, so we get an $n$-cube.)

Does anyone know exactly where this parametrization of the cubical structure is written down? I know I cannot eliminate "guess" responses, but let me try by saying that I have checked all the reasonable guesses and have not found it. I know it is out there. Does anyone know where?