Hi everyone. My question is about the absolute Galois group action on the set of the Grothendieck dessins. The dessins I am interested in are trees with only one vertex of valency more then 2. (I don't know if there is a generally accepted term in graph theory. Starlike trees?). What exactly is known about them? Is the action transitive, at least for trees with 3 terminal vertices (with the same valency lists)?
EDIT: I see some clarification is necessary. In fact, I consider the alternating dessins, so there are two valency lists for each of them. (It is convenient to assume that the "center" is always, say, black). As Will Savin pointed out, a dessin with rotational symmetry cannot turn into one with less symmetry. I confess I missed this obvious fact, but certainly this is not what I had in mind. The question was actually about nontrivial obstacles to transitivity. (I know there are some for general trees).
Then, "what is known" part of the question. The best answer would be a long list of references.