The trees with the second smallest normalized Laplacian eigenvalue at least 1−32

Let λ2(G) be the second smallest normalized Laplacian eigenvalue of a graph G. In this paper, we determine all trees with λ2(T)≥1−32. Meanwhile, if T is a tree of diameter 6, then we prove that λ2(T)≤1−32. Moreover, we determine all trees of diameter d=3 or 6 with λ2(T)=1−32.