  We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in
the exterior unit disc, and the image of the unit circle under $f$ has the
maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a
bi-angled tree associated to any such $f$. It is proven that any bi-angled tree
is realizable by such an $f$, and moreover, $f$ is essentially uniquely
determined by its associated bi-angled tree. This combinatorial classification
is used to show that such $f$ are in natural 1:1 correspondence with
anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical
points (classified by their Hubbard trees).
