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In recent years, there has been an increasing interest in peer-to-peer (P2P) multimedia streaming. In this paper, we consider constructing a high-bandwidth overlay tree for streaming services. We observe that underlay information such as link connectivity and link bandwidth is important in tree construction, because two seemingly disjoint overlay paths may share common links on the underlay. We hence study how to construct a high-bandwidth overlay tree given the underlay topology. We formulate the problem as building a Maximum Bandwidth Multicast Tree (MBMT) or a Minimum Stress Multicast Tree (MSMT), depending on whether link bandwidth is available or not. We prove that both problems are NP-hard and are not ap-proximable within a factor of (2/3 + epsiv), for any epsiv > 0, unless P = NP. We then present approximation algorithms to address them and analyze the algorithm performance. Furthermore, we discuss some practical issues (e.g., group dynamics, resilience and scalability) in system implementation. We evaluate our algorithms on Internet-like topologies. The results show that our algorithms can achieve high tree bandwidth and low link stress with low penalty in end-to-end delay. Measurement study based on Plan-etLab further confirms this. Our study shows that the knowledge of underlay is important for constructing efficient overlay trees.