Upper bound of the multiplicity of Laplacian eigenvalue 1 of trees

Let T be a tree of order $n(\ge 6)$. The number of pendant vertices (resp., quasi-pendant vertices) of T is denoted by $p\left(T\right)$ (resp., $q\left(T\right)$). Let $m_{L\left(T\right)}\left(\lambda \right)$ denote the multiplicity of λ as a Laplacian eigenvalue of T. The multiplicity of 1 as a Laplacian eigenvalue of T has attracted much attention. In this paper, we first prove that$m_{L\left(T\right)}\left(1\right)=p\left(T\right)-q\left(T\right)+m_{L\left(\bar{T}\right)}\left(1\right),$ where $\bar{T}$ is called the reduced tree of T, obtained from T by deleting some pendant vertices such that $p\left(\bar{T}\right)=q\left(\bar{T}\right)$. Further, for each reduced tree T of order $n(\ge 6)$, we prove that$m_{L\left(T\right)}\left(1\right)\le \frac{n-2}{4},$ and the structure of the extremal trees attaining the upper bound is characterized completely.