Cospectral trees indistinguishable by scattering

Let v_1 and v_2 be two distinct vertices of a tree T_0. Let φ_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let φ_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then φ_N(λ)^{(1)}\equiv φ_N(λ)^{(2)} and φ_D(λ)^{(1)}\equiv φ_D(λ)^{(1)} (which means that the scattering is the same at v_1 and v_2).