  We describe a class of rank-2 graphs whose C^*-algebras are AT algebras. For
a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of
the C*-algebra. We identify rank-2 Bratteli diagrams whose C*-algebras are
simple and have real-rank zero, and characterise the K-invariants achieved by
such algebras. We give examples of rank-2 Bratteli diagrams whose C*-algebras
contain as full corners the irrational rotation algebras and the Bunce-Deddens
algebras.
