Summary: Let \(G\) be a graph with adjacency matrix \(A(G)\) and let \(D(G)\) be the diagonal matrix of the degrees of \(G\). For every real \(\alpha \in [0, 1]\), define the matrix \(A_\alpha(G)\) as \[ A_\alpha(G) = \alpha D(G) +(1 - \alpha) A(G) . \] This paper gives several results about the \(A_\alpha\)-matrices of trees. In particular, it is shown that if \(T_{\operatorname{\Delta}}\) is a tree of maximal degree {\(\Delta\)}, then the spectral radius of \(A_\alpha(T_{\operatorname{\Delta}})\) satisfies the tight inequality \[ \rho(A_\alpha(T_{\operatorname{\Delta}})) < \alpha \operatorname{\Delta} + 2(1 - \alpha) \sqrt{\operatorname{\Delta} - 1}, \] which implies previous bounds of C. D. Godsil [Ann. Discrete Math. 20, 151–159 (1984; Zbl 0559.05040)], L. Lovász [Combinatorial problems and exercises. 2. ed. Amsterdam: North-Holland. (1993; Zbl 0785.05001)], and D. Stevanović [Linear Algebra Appl. 360, 35–42 (2003; Zbl 1028.05062)]. The proof is deduced from some new results about the \(A_\alpha\)-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of \(A_\alpha\) of general graphs are proved, implying tight bounds for paths and Bethe trees.