  Hypertrees and noncrossing trees are well-established objects in the
combinatorics literature, but the hybrid notion of a noncrossing hypertree has
received less attention. In this article I investigate the poset of noncrossing
hypertrees as an induced subposet of the hypertree poset. Its dual is the face
poset of a simplicial complex, one that can be identified with a generalized
cluster complex of type $A$. The first main result is that this noncrossing
hypertree complex is homeomorphic to a piecewise spherical complex associated
with the noncrossing partition lattice and thus it has a natural metric. The
fact that the order complex of the noncrossing partition lattice with its
bounding elements removed is homeomorphic to a generalized cluster complex was
not previously known or conjectured.
  The metric noncrossing hypertree complex is a union of unit spheres with a
number of remarkable properties: 1) the metric subspheres and simplices in each
dimension are both bijectively labeled by the set of noncrossing hypertrees
with a fixed number of hyperedges, 2) the number of spheres containing the
simplex labeled by the noncrossing tree $\tau$ is the same as the number
simplices in the sphere labeled by the noncrossing tree $\tau$, and 3) among
the maximal spherical subcomplexes one finds every normal fan of a metric
realization of the simple associahedron associated to the cluster algebra of
type $A$. In particular, the poset of noncrossing hypertrees and its metric
simplicial complex provide a new perspective on familiar combinatorial objects
and a common context in which to view the known bijections between noncrossing
partitions and the vertices/facets of simple/simplicial associahedra.
