Generalized spectral characterization of signed trees

Let $T$ be a tree with an irreducible characteristic polynomial $\phi (x)$ over $\mathbb{Q}$. Let $\mathrm{\Delta }(T)$ be the discriminant of $\phi (x)$. It is proved that if $2^{-\frac{n}{2}}\sqrt{\mathrm{\Delta }(T)}$ (which is always an integer) is odd and square free, then every signed tree with underlying graph $T$ is determined by its generalized spectrum.