  We construct a bijection between certain Deodhar components of a braid
variety constructed from an affine Kac-Moody group of type $A_{n-1}$ and
vertex-labeled trees on $n$ vertices. By an argument of Galashin, Lam, and
Williams using Opdam's trace formula in the affine Hecke algebra and an
identity due to Haglund, we obtain an elaborate new proof for the enumeration
of the number of vertex-labeled trees on $n$ vertices.
