  The paper has three parts. In the first part we apply the theory of commuting
pairs of (pseudo) difference operators to the (formal) asymptotics of
orthogonal polynomials: using purely geometrical arguments we show
heuristically that the asymptotics, for large degrees, of orthogonal polynomial
with respect to varying weights is intimately related to certain spinor bundles
on a hyperelliptic algebraic curve reproducing formulae appearing in the works
of Deift et al. on the subject.
  In the second part we show that given an arbitrary nodal hyperelliptic curve
satisfying certain conditions of admissibility we can reconstruct a sequence of
polynomials orthogonal with respect to semiclassical complex varying weights
supported on several curves in the complex plane. The strong asymptotics of
these polynomials will be shown to be given by the spinors introduced in the
first part using a Riemann-Hilbert analysis.
  In the third part we use Strebel theory of quadratic differentials and the
procedure of welding to reconstruct arbitrary admissible hyperelliptic curves.
As a result we can obtain orthogonal polynomials whose zeroes may become dense
on a collection of Jordan arcs forming an arbitrary forest of trivalent
loop-free trees.
