Shabat polynomials and monodromy groups of trees uniquely determined by ramification type

A dessin d’enfant or dessin is a bicolored graph embedded into a Riemann surface. Acyclic dessins can be described analytically by preimages of Shabat polynomials and algebraically by their monodromy groups. We determine the Shabat polynomials and monodromy groups of planar acyclic dessins that are uniquely determined by their ramification types.