Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled quadratic polynomials, or forms, in two variables.

The refined face partition polynomials of the permutohedra (OEIS A049019) and associahedra (or Stasheff polytopes, A133437) and partition polynomials for noncrossing partitions (A134264) can all be algebraically related by re-scaling the associated monomials per the calculus of compositional and multiplicative inversion of power series, so this represents the algebraic analogue of the rescaling of the quadratic equations for conics.

Which geometric structures, such as possibly the noncrossing hypertrees of McCammond, brick polytopes, or simply polygons, when "sliced" in different ways, provide representations of permutohedra, associahedra, and noncrossing partitions?

Ancillary question: Is there a geometric complex (possibly the Whitehouse simplicial complex, related to phylogenetic trees, or the trees themselves) that can be associated to Lagrange (compositional) inversion of exponential generating functions (i.e., formal Taylor, or divided powers, series) and can be incorporated in this geometric scheme if one exists?