Let　Delta(T)　and　mu(T)　denote　the　maximum　degree　and　the　Laplacian　spectral　radius　of　a　tree　T.　respectively.　In　this　paper　we　prove　that　for　two　trees　T-1　and　T-2　on　n　(n　＞=　21)　vertices,　if　Delta(T-1)　＞　Delta(T-2)　and　Delta(T-1)　＞=　inverted　right　perpendicular11n/30inverted　left　perpendicular　+　1,　then　mu(T-1)　＞　mu(T-2),　and　the　bound　＂Delta(T-1)　＞=　inverted　right　perpendicular11n/30inverted　left　perpendicular　+　1＂　is　the　best　possible.　We　also　prove　that　for　two　trees　T-1　and　T-2　on　2k　(k　＞=　4)　vertices　with　perfect　matchings,　if　Delta(T-1)　＞　Delta(T-2)　and　Delta(T-1)　＞=　inverted　right　perpendiculark/2inverted　left　perpendicular　+　2,　then　mu(T-1)　＞　mu(T-2)　(C)　2011　Elsevier　BM.　All　rights　reserved.