Although many overlay and P2P approaches have been proposed to assist large-scale live video streaming, how to ensure service quality and reliability still remains a challenging issue. Peer dynamics, especially unscheduled node departures, affect the perceived video quality at a peer node in two ways. In particular, the amplitude of quality fluctuations and the duration for which stable quality video is available at a node heavily depend on the nature of peer departures in the system. In this paper, we first propose a service quality model to quantify the quality and stability of a video stream in a P2P streaming environment. Based on this model, we further develop tree construction algorithms that ensure that every peer in the collaborative network receives a video stream with a statistical reliability guarantee on quality. A key advantage of the proposed approach is that we can now explicitly control the quality and stability of the video stream supplied to every node. This is the fundamental difference of our approach from existing approaches that provide stream stability by over-provisioning resources allocated to every peer. Also, the proposed tree construction schemes decide the position of a node in the delivery tree based on both its estimated reliability and upstream bandwidth contribution while striving to minimize the overall load on the server. Our simulations show that our algorithms use the server resources very efficiently while significantly improving the video stability at peers.