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Network coding is known to provide increased throughput and reduced delay for communications over networks. In this paper we propose a peer-to-peer video streaming system that exploits network coding in order to achieve low start-up delay, high streaming rate, and high resiliency to peers' dynamics. In particular, we introduce the concept of random multicast trees as overlay topology. This topology offers all benefits of tree-based overlays, notably a short start-up delay, but is much more efficient at distributing data and recovering from ungraceful peers departures. We develop a push-based streaming system that leverages network coding to efficiently distribute the information in the overlay without using buffer maps. We show performance results of the proposed system and compare it with an optimized pull systems based on Coolstreaming, showing significant improvement.