Towards tropical dessins d'enfants

There are several mathematical worlds inhabited by graphs with some additional structure. We are best acquanted with dessins d'enfants that are closely related to the metrized ribbon graphs. Allowing infinite lengths leads to the world of tropical curves and their moduli, which are connected to the classical moduli spaces by means of Berkovich non-archimedean analytic spaces.
We are going to summarize the connections between the above structures established in the recent papers by several authors and suggest our own ones, mostly conjectural. Several examples will be presented.