Let Δ(T) and λ 1 (T) denote the maximum degree and the largest eigenvalue of a tree T, respectively. Let Tn be the set of trees on n vertices, and Tn(Δ)={T∈Tn|Δ(T)=Δ}. In the present paper, among the trees in Tn(Δ) (n⩾4), we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when n-22⩽Δ⩽n-1. Furthermore, it is proved that, for two trees T 1 and T 2 in Tn (n⩾4), if Δ(T1)⩾2n3-1 and Δ(T 1 )>Δ(T 2 ), then λ 1 (T 1 )>λ 1 (T 2 ). By applying this result, we extend the order of trees in Tn by their largest eigenvalues to the 13th tree when n⩾12. This extends the results of Hofmeister [Linear Algebra Appl. 260 (1997) 43] and Chang et al. [Linear Algebra Appl. 370 (2003) 175].