Eigenvalue monotonicity of $q$-Laplacians of trees along a poset

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$. Let $GT{S}_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q\in R$, going up on $GT{S}_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_T^q$ increase while the smallest eigenvalue of $L_T^q$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.