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We consider Sturm-Liouville operators on geometrical graphs without cycles(trees) with singular potentials from the class $W_2^{-1}$. We suppose that thepotentials are known on a part of the graph, and study the so-called partialinverse problem, which consists in recovering the potentials on the remainingpart of the graph from some parts of several spectra. The main results of thepaper are the uniqueness theorem and a constructive procedure for the solutionof the partial inverse problem. Our method is based on the completeness and theRiesz-basis property of special systems of vector functions, and the reductionof the partial inverse problem to the complete one on a part of the graph.

An inverse problem for Sturm-Liouville operators on trees with partial information given on the potentials