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In this paper, we study the problem of utility maximization in peer-to-peer (P2P) systems, in which aggregate application-specific utilities are maximized by running distributed algorithms on P2P nodes, which are constrained by their uplink capacities. For certain P2P topologies, we show that routing along a linear number of trees per source can achieve the largest rate region that can be possibly obtained by intrasession and intersession network coding. This observation allows us to develop a simple multitree formulation for the problem. For the resulting nonstrictly concave optimization problem, we develop a Primal-dual distributed algorithm and prove its global convergence using our proposed sufficient conditions. These conditions are general and add understanding to the convergence of primal-dual algorithms under nonstrictly concave settings. We implement the proposed distributed algorithm in a peer-assisted multiparty conferencing system by utilizing only end-to-end delay measurements between P2P nodes. We demonstrate its superior performance through actual experiments on a LAN testbed and the Internet.