A note on multiplicity of the Laplacian eigenvalue of trees

Considering the multiplicity m
T
(λ) of eigenvalue λ (which equals 1) of Laplacian matrix of all trees, we get three results: When m
T
(1) equals n-2, the tree is unique, that is star graph K
1,n-1
; (ii) there exists no trees satisfying m
T
(1) equals n-3; (iii) When m
T
(1) equals n-4, this kind of trees are divided into two types. According to the process of proving, we devise a method to construct trees on some desired properties, which have practical value.