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Recently, peer-to-peer video streaming technology has been widely suggested to support broadcasting videos over the Internet. Given that the application-layer nodes (peers) are autonomous and may join or leave at will. Dealing with node dynamics (known as churn) under the bandwidth and timing constraints for video streaming thus becomes a necessary yet challenging task. One conventional argument is that the stable nodes are too small a group to be effectively utilized. While we agree this argument from a whole session's point-of-view, in this paper, we argue that the significantly longer lifespans of the stable nodes allow each of them to appear in much more snapshots of the overlay than transient nodes and, as a result, they constitute a significant portion in every snapshot of the overlay, substantially affecting the performance of the overall system. Such observation has been verified by our trace analysis on PPLive, a large-scale peer-to-peer live streaming system, as well as analytical modeling based on node behaviors. Motivated by this, we further propose a tiered overlay design, with stable nodes being organized into a tier-1 backbone for serving tier-2 nodes. It offers a highly cost-effective and deployable alternative to proxy assisted designs. We develop a comprehensive set of algorithms for stable node identification and organization. Specifically, we present a novel structure, labeled tree (or LBTree in short), for the tier-1 overlay, which, leveraging stable peers, simultaneously achieves low overhead and high transmission reliability. Our tiered framework flexibly accommodates diverse existing overlay structures in the second tier. Through extensive simulations, we demonstrate that the customized optimization using selected stable nodes can greatly boost the streaming quality and also effectively reduce the control overhead.