This paper studies the average tree solution with fuzzy communication structure. In fuzzy graphs, the membership of the players and the relations among them are leveled. This paper introduces Choquet by graphs version on fuzzy communication structure, whereby the fuzzy average tree solution is defined, which is a new solution that considers the bilateral communications as players. On the class of cycle-free fuzzy graph games, the fuzzy average tree solution is the unique solution that satisfies fuzzy component efficiency and cg component fairness as well as the unique solution that satisfies fuzzy component efficiency and fuzzy component fairness. Especially, it is proved that average tree solution is an element of the core for superadditive games which illustrates stability. We extend and generalize the average tree solution using the classical DFS algorithm from games with cycle-free communication structure to the class of all games with communication structure, and finally calculate the fuzzy average tree solution in a supply chain example.