This paper focuses on the consensus problems of multiagent systems with linear and nonlinear dynamics for both leader and leaderless cases. In order to overcome the drawback that the design of control protocols depends on the global information of the network in previous works, a novel distributed consensus protocol, updating the weights of the directed spanning tree of the communication network (directed spanning tree based adaptive control protocol), is developed. A scheme for reordering the nodes is proposed, and by using this scheme, an equivalent lemma for achieving consensus is given. Applying the developed method and Lyapunov stability theory, some distributed adaptive laws are designed on the coupling weights in a directed network. It is found that the consensus can be reached by randomly choosing a directed spanning tree and using the developed distributed adaptive laws. Furthermore, by reordering the nodes and using the proposed adaptive laws, one obtains that the leader-following consensus can be reached by pinning a small fraction of nodes. For both, with the leader and leaderless cases, it is also found that the designed adaptive law on the weight of the directed spanning tree depends on the position of the head node of the directed edge in the directed spanning tree. Finally, three examples are presented to illustrate the theoretical analysis.