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This paper considers the problem of power-efficient distributed estimation of vector parameters related to localized phenomena so that both the subset of sensor selection and the routing structure in a wireless sensor network are optimized jointly in order to obtain the best possible estimation performance at a given querying node, for a given total power budget. We first formulate our problem as an optimization problem and show that it is NP-Hard. Then, we design two algorithms: a fixed-tree relaxation-based and a novel and very efficient local distributed optimization to optimize jointly the sensor selection and the routing structure. We also provide a lower bound for our optimization problem and show that our local distributed optimization algorithm provides a performance that is close to this bound. Although there is no guarantee that the gap between this lower bound and the optimal solution of the main problem is always small, our numerical experiments support that this gap is actually very small in many cases. An important result from our work is that because of the interplay between the communication cost over the links and the gains in estimation accuracy obtained by choosing certain sensors, the traditional shortest-path-tree routing structure, widely used in practice, is no longer optimal, that is, our routing structures provide a better trade-off between the overall power efficiency and the final estimation accuracy obtained at the querying node. Comparing to more conventional sensor selection and fixed routing algorithms, our proposed joint sensor selection and routing algorithms yields a significant amount of energy saving.