  We associate a complete non-singular fan with a polygon triangulation. Such a
fan appears from a certain toric Richardson variety, called of Catalan type
introduced in this paper. A toric Richardson variety of Catalan type is a Fano
Bott manifold. We show that toric Richardson varieties of Catalan type are
classified up to isomorphism in terms of unordered binary trees. In particular,
the number of isomorphism classes of $n$-dimensional toric Richardson varieties
of Catalan type is the $(n+1)$th Wedderburn--Etherington number.
