  We solve two inverse spectral problems for star graphs of Stieltjes strings
with Dirichlet and Neumann boundary conditions, respectively, at a selected
vertex called root. The root is either the central vertex or, in the more
challenging problem, a pendant vertex of the star graph. At all other pendant
vertices Dirichlet conditions are imposed; at the central vertex, at which a
mass may be placed, continuity and Kirchhoff conditions are assumed. We derive
conditions on two sets of real numbers to be the spectra of the above Dirichlet
and Neumann problems. Our solution for the inverse problems is constructive: we
establish algorithms to recover the mass distribution on the star graph (i.e.
the point masses and lengths of subintervals between them) from these two
spectra and from the lengths of the separate strings. If the root is a pendant
vertex, the two spectra uniquely determine the parameters on the main string
(i.e. the string incident to the root) if the length of the main string is
known. The mass distribution on the other edges need not be unique; the reason
for this is the non-uniqueness caused by the non-strict interlacing of the
given data in the case when the root is the central vertex. Finally, we relate
of our results to tree-patterned matrix inverse problems.
