Proof of a problem on Laplacian eigenvalues of trees

Denote by μk(L(T))μk(L(T)) the k-th Laplacian eigenvalue of a tree T  . Let Tk(2t)Tk(2t) be the set of all trees of order 2tk2tk with perfect matchings. In this note, the trees T   in Tk(2t)Tk(2t) with μk(L(T))=t+2+t2+42 are characterized, which solves Problem of J.X. Li, W.C. Shiu and A. Chang in [3] completely.