  A k-triangulation of a convex polygon is a maximal set of diagonals so that
no k+1 of them mutually cross in their interiors. We present a bijection
between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths
of length 2(n-4). This solves the problem of finding a bijective proof of a
result of Jonsson for the case k=2. We obtain the bijection by constructing
isomorphic generating trees for the sets of 2-triangulations and pairs of
non-crossing Dyck paths.
