The client-server architecture widely adopted on the Internet is not adequate to meet the ever-increasing user loads and bandwidth demands in live streaming systems especially for multimedia content delivery. Peer-to-peer (P2P) overlay networks provide excellent system scalability and high resource utilization, which make it an attractive solution to this problem. We consider a hybrid hierarchical P2P overlay network that consists of both super and normal peers to support live streaming applications. This architecture is built upon a tree-structured network of super peers, which organize normal peers into clusters. The tree construction process has a significant impact on the overall system performance. We formulate a specific type of problem, max-minTC, to maximize the minimum node throughput in tree construction, where the system's stream rate is optimized by constructing an efficient spanning tree among super peers. We present a decentralized approach where super peers run the same algorithm in parallel to derive a tree from an identical database describing the topology of the streaming system. This approach is able to quickly converge to a new tree upon the detection of any topological changes in super peers. The performance superiority of the proposed solution is illustrated by extensive simulations on a large set of simulated networks of various sizes from small to large scales in comparison with other methods.