On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem

Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. It is proved that there are no co-spectral (i.e., having the same spectrum of such problem) among equilateral trees of $\leq 8$ vertices. All co-spectral trees of $9$ vertices are presented.