  The inverse problem of recovery of a potential on a quantum tree graph from
Weyl's matrix given at a number of points is considered. A method for its
numerical solution is proposed. The overall approach is based on the leaf
peeling method combined with Neumann series of Bessel functions (NSBF)
representations for solutions of Sturm-Liouville equations. In each step, the
solution of the arising inverse problems reduces to dealing with the NSBF
coefficients. The leaf peeling method allows one to localize the general
inverse problem to local problems on sheaves, while the approach based on the
NSBF representations leads to splitting the local problems into two-spectra
inverse problems on separate edges and reduce them to systems of linear
algebraic equations for the NSBF coefficients. Moreover, the potential on each
edge is recovered from the very first NSBF coefficient. The proposed method
leads to an efficient numerical algorithm that is illustrated by numerical
tests.
