Ordering trees by their largest eigenvalues

The set of trees with n vertices is denoted by . Hofmeister has determined the first five values of the largest eigenvalue of trees in and the corresponding trees for these values [Linear Algebra Appl. 260 (1997) 43]. In other words, an order of the first five trees in by their largest eigenvalues has been given. Focus on the same purpose, we shall give a partition for trees in first in the this paper and then extend this order to the eighth tree.