On the inverse problem of the two-velocity tree-like graph

In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincar{\'e} operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.