Per_n is an affine algebraic curve, defined over Q, parametrizing (up to change of coordinates) degree-2 self-morphisms of P^1 with an n-periodic ramification point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of C with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is Per_n is irreducible over C? (2) Is G_n is irreducible over Q?

We show that if G_n is irreducible over Q, then Per_n is irreducible over C. In order to do this, we find a Q-rational smooth point of a projective completion of Per_n. This Q-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits a tropical interpretation.