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In this technical note we consider the problem of distributed discrete-time state estimation over sensor networks. Given a graph that represents the sensor communications, we derive the optimal estimation algorithm at each sensor. We further provide a closed-form expression for the steady-state error covariance matrices when the communication graph reduces to a directed tree. We then apply the developed theoretical tools to compare the performance of two sensor trees and convert a random packet-delay model to a random packet-dropping model. Examples are provided throughout the technical note to support the theory.