Trees with maximum sum of the two largest Laplacian eigenvalues

Let $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher et al. proved that for any tree $T$ of order $n$, $S_2(T) \leq n+2-\frac{2}{n}$. Guan et al. determined the tree with maximum $S_2(T)$ among all trees of order $n$. In this paper, we characterize the trees with $S_2(T) \geq n+1$ among all trees of order $n$ except some trees. Moreover, among all trees of order $n$, we also determine the first $\lfloor\frac{n-2}{2}\rfloor$ trees according to their $S_2(T)$. This extends the result of Guan et al.