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Optimal power allocation for distributed parameter estimation in a wireless sensor network with a fusion center under a total network power constraint is considered. For the simple star topology, an analysis of the effect of the measurement noise variance on the optimal power allocation policy is presented. The optimal solution evolves from sensor selection, to water- filling, to channel inversion as the measurement noise variance increases; in the last solution, the sensor with the weakest channel signal-to-noise ratio (SNR) is allocated the largest fraction of the total power. Relaying nodes are then introduced to form the more complex branch, tree, and linear topologies. The optimal power allocation strategies for these complex topologies are then considered for both amplify-and-forward and estimate-and-forward transmission protocols. Analytical solutions for these cases appear to be intractable, and thus asymptotically optimal (for increasing measurement noise variance) solutions are derived. The solutions to this asymptotic problem offer near-optimal performance even for modest measurement noise. The optimal limiting power policy for the leaf nodes in branch and tree topologies is channel inversion, whereas in linear networks, the optimal solution is a form of weighted channel inversion. The results are extended to a multipath channel model and to the estimation of a vector of random parameters.