Hecke triangle Groups and Dessin d'enfant

In this work we will construct bipartite graphs, famously known as Dessin d'enfant, corresponding to finite index subgroups of Hecke triangle groups $(2, q, \infty )$. Then using a results of \cite{ll} we shall show the correspondences among the special polygons, the bi-partite graph, and the tree diagram for a finite index subgroup of the Hecke triangle groups $(2, q, \infty )$.