New answers tagged motivation
For dessin d'enfants, I believe the orientation is superfluous - each edge goes between a black vertex and a white vertex, so picking an orientation is just picking one of those, whcih doesn't help uniformize anything. A dessin d'enfants corresponds to a cover of $\mathbb P^1$ ramified over three points $(0,1,\infty)$. A rooted dessin d'enfants, I think, ...
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