  We give a recursive formula, expressed in terms of the characteristic tuples,
for the Betti numbers of the boundary of the Milnor fiber of an irreducible
quasi-ordinary surface. The singular locus of the surface consists of two
components, and for each component we introduce a sequence of increasingly
simpler surfaces. Our recursion depends on a detailed comparison of these two
sequences. We indicate how we expect pieces of these associated surfaces to
glue together to reconstruct the Milnor fiber and its boundary.
