  In this paper, we study a class of toric ideals obtained by using some
geometric data of ADE trees which are the minimal resolution graphs of rational
surface singularities. We compute explicit Gr\"obner bases for these toric
ideals that are also minimal generating sets consisting of large number of
binomials of degree $\leq 4$. In particular, they give rise to squarefree
initial ideals as well.
