This paper develops a novel triangular fuzzy noncooperative-cooperative biform game framework to examine the formation of a bilateral link network and addresses profit allocations with triangular fuzzy coalition characteristic values. This triangular fuzzy noncooperative-cooperative biform game consists of four elements: nodes (or players), link modes (or strategy choice), network formation (or coalitions) and information flow (or triangular fuzzy coalition characteristic functions), and contains both the triangular fuzzy noncooperative and cooperative components. The payoffs are not directly given a priori in the triangular fuzzy noncooperative part, but are determined by solving the triangular fuzzy cooperative part. To solve the triangular fuzzy cooperative games, we redefine the Banzhaf values under the triangular fuzzy numbers case. In addition, we establish the general existence conditions of the Nash equilibrium in the triangular fuzzy noncooperative-cooperative biform game. Finally, numerical example is used to verify the effectiveness, applicability and complexity of the proposed models and methods, and this framework provides a new way to solve the problems of both strategy choices and profits allocation.