Faithful action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on dessins of genus $0$

Question

In a nearly classic paper of '94, L. Schneps recalls a proof by Lenstra of the fact that the absolute Galois group acts faithfully on Shabat polynomials (i.e. on trees). As immediate corollary, she has that the absolute Galois group acts faithfully on dessins of genus $0$. I don't understand why it suffices to give the proof for trees in order to show the claim for general dessins. In fact, a dessin of genus $0$ could have more than one face, unlike trees that have only one face. Moreover, a "multifaced" (having more than one face) dessin is not isomorphic to a tree. So, how can she deduce the faithfulness of the action on the whole collection of dessins of genus $0$ through Lenstra's theorem?

Answer 1

This is because Shabat polynomials are a subset of genus $0$ dessins, so the action being faithful on them implies it being faithful on genus $0$ dessins. (Recall that this last statement means that for any $\sigma \in \text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ other than the identity, there is a genus $0$ dessin $\mathcal D$ such that $\mathcal D^\sigma \ncong \mathcal D$. Since there is such a genus $0$ tree by Lenstra's theorem, this holds).