There is a combinatorial formula for the number of geometric genus g curves of degree d (possibly reducible) passing through 3d−1+g generic points of mathbbP2 (or a more general toric surface) derived by tropical techniques, see
Grigory Mikhalkin, Enumerative tropical algebraic geometry in mathbbR2,
J. Amer. Math. Soc. 18 (2005), no. 2, 313–377 MR
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