Due to churn as well as node and link failures in peer-to-peer (P2P) networks, providing resilient multicast is a challenging issue, particularly for overlay trees. In this work, we first study analytical properties of the backup parent pool that aims at improving the resilience of overlay multicast trees in P2P video streaming. We then present a novel greedy degree-constrained multicast tree construction algorithm that addresses the tradeoffs between maximizing resiliency, maximizing bandwidth utilization, and minimizing delay. The choice of essential design parameters is studied together with seamlessness of the streaming under a variety of fault scenarios. Simulation results indicate that the overhead introduced by the mechanism is negligible if there is sufficient bandwidth in the system. Both analytic and simulation results indicate that the proposed approach improves resiliency, bandwidth utilization, and delay. Comparisons with an algorithm from the literature are carried out and it is observed that the proposed method is superior under a wide range of conditions.