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Consider a network which can be represented by an acyclic directed graph such that the links represented by the edges are subject to failure. Under the assumption that at most one link can fail at any time, we want to locate a failed link, if any, by means of certain tests. A test is performed by injecting a signal at a vertex and monitoring it at another vertex and can reveal if there is a failed link on any path between the two vertices. We want to find a minimal set of tests that can uniquely locate any single fault. Since this problem is in general NP-complete, we investigate a special case where the given network has a tree structure. We present an algorithm whose worst case running time can be bounded by a linear function of the input size.