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Graphical models provide a powerful general framework for encoding the structure of large-scale estimation problems. However, the graphs describing typical real-world phenomena contain many cycles, making direct estimation procedures prohibitively costly. In this paper, we develop an iterative inference algorithm for general Gaussian graphical models. It operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph. When these subproblems are suitably chosen, the algorithm converges to the correct conditional means. Moreover, and in contrast to many other iterative methods, the tree-based procedures we propose can also be used to calculate exact error variances. Although the conditional mean iteration is effective for quite densely connected graphical models, the error variance computation is most efficient for sparser graphs. In this context, we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree-structured counterparts, thanks both to the expressive power of these models and to the efficient inference algorithms developed herein. The convergence properties of the proposed tree-based iterations are characterized both analytically and experimentally. In addition, by using the basic tree-based iteration to precondition the conjugate gradient method, we develop an alternative, accelerated iteration that is finitely convergent. Simulation results are presented that demonstrate this algorithm's effectiveness on several inference problems, including a prototype distributed sensing application.