The tree algorithm, a new basic protocol for accessing a satellite channel by a large number of independent users, is considered. Previous multi-accessing techniques suffered from long delays, low throughput and/or congestion instabilities. The ALOHA algorithm, for example, when applied to the Poisson source model, has a maximum throughput of .37 packets/slot and is unstable. The tree protocol, under similar conditions, is stable, has a maximum average throughput of .43 and has respectable delay properties. In this protocol, the sources are assigned to the leaves of a tree graph. Message contentions are resolved by systematically moving from node to node through the tree, trying to determine the branches containing the conflicting users. It is shown that the tree protocol is a generalization of TDMA and that an optimum dynamic tree adaptively changes from an essentially random access protocol in light traffic to TDMA in heavy traffic. A major result is that ifq. the probability that a user has a packet to transmit, is greater than1/\sqrt{2}, then TDMA and the optimum tree protocol axe the same. If, on the other hand,q < 1/\sqrt{2}, then the optimum tree protocol is more efficient than TDMA. The maximum average throughput of a system consisting of the optimum tree protocol and a finite number of users is one packet/slot. The tree protocol may be applied either to a direct access system or to a demand assignment access system. Consequently, an example is presented comparing the average delay properties of tree and TDMA protocols when applied to direct access and to reservation access systems.