Conservative polynomials and yet another action of $\Gal(\bar \Q/\Q)$ on plane trees

In this paper we study an action of the absolute Galois group $\Gamma$ on bicolored plane trees induced by the action of $\Gamma$ on equivalence classes of conservative polynomials which are the simplest example of postcritically finite rational functions. We establish some basic properties of this action and compare it with the similar action provided by the Grothendieck theory of "Dessins d'enfants".