Multipoint communication has been an increasingly focused topic in computer communication networks, including the Internet, the ATM, and the wireless/mobile networks. The major challenges of designing multicast flow control protocols for a combined wired/wireless network are the varying transmission characteristics (bandwidth, error, reliability, and propagation delay) of the wireless and wired media, and the irregular, different, possibly conflicting flow control requests from multiple receivers in the point-to-multipoint (branching) setting, or from multiple senders in the multipoint-to-point (merging) setting. To address these issues, in this paper we design, analyze, and evaluate both branch-point and merge-point algorithms; we also provide a comparison on the major issues and solutions of the two multicasting scenarios. On point-to-multipoint ABR flow control, we examine an existing max–min fair branch-point algorithm proposed by Siu and Tzeng, and formally analyze its maximum cell loss. A new algorithm is then proposed. Both the maximum cell loss and max–min fairness of new algorithm are analyzed. With extensive simulation, we compare three branch-point algorithms (including a third one proposed by Fahmy, Jain et al.). On multipoint-to-point flow control, we extend the “essential fairness” concept, which was first proposed by Wang and Schwartz to flow control of multicast and unicast TCP traffic in the Internet to the multipoint-to-point ABR flow control. We design a general switch algorithm, which provides essential fairness to the multipoint-to-point ABR flow control. Three major variations of the algorithm, each of them guarantees different fairness (of favor) to unicast or multicast sessions, are then presented. These three schemes are evaluated by simulation. The significance of our approach is illustrated by the formal analysis of cell loss, feedback delay and max–min fairness properties, the generalized fairness definition for multicasting flow control, and the comparisons we made for (1) point-to-multipoint vs. multipoint-to-point scenarios, (2) three branch-point algorithms, and (3) three fairness definitions and mechanisms of multipoint-to-point multicasting. The work may be applied to multicast services in various high speed networks, such as supporting differentiated multicast service over IP, supporting flexible billing schemes, and providing QoS services over hybrid networks.