Bistability, i.e., the existence of just two stable equilibria, is known to play an important role in biological systems, e.g., cellular differentiation and apoptosis. In this paper, we consider the bistability but as a structural property of a class of network systems, that is, the bistability under the assumption that the information on the network structure is available but the information on the components is not available. First, we introduce Boolean networks as a model of biological network systems and give the notion of structural bistability. We next focus on the systems with a flower-shaped network structure and present a necessary and sufficient condition based on three characteristics of the network topology. Finally, the result is extended to the Boolean networks with a chain-shaped network structure.