  We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial
formulas for Severi degrees.
  The degree of the Severi variety of plane curves of degree d and delta nodes
is given by a polynomial in d, provided delta is fixed and d is large enough.
We extend this result to generalized Severi varieties parametrizing plane
curves which, in addition, satisfy tangency conditions of given orders with
respect to a given line. We show that the degrees of these varieties,
appropriately rescaled, are given by a combinatorially defined "relative node
polynomial" in the tangency orders, provided the latter are large enough. We
describe a method to compute these polynomials for arbitrary delta, and use it
to present explicit formulas for delta <= 6. We also give a threshold for
polynomiality, and compute the first few leading terms for any delta.
