A new characterization of the set of Laplacian spectral radii of trees

For any positive integer $r$ and real number $\alpha >1$, let ${\mathcal{L}}_r(\alpha )$ denote the set of positive real numbers defined recursively: $\alpha -1\in {\mathcal{L}}_r(\alpha )$, and for any multi-subset $\{q_1,q_2,\dots ,q_s\}$ of ${\mathcal{L}}_r(\alpha )$, where $0<s<r$, $\beta :=\alpha -1-s-{\sum }_{i=1}^s{q}_i^{-1}$ belongs to ${\mathcal{L}}_r(\alpha )$ as long as $\beta >0$. We show that $(\alpha -1{)}^{-1}\in {\mathcal{L}}_r(\alpha )$ if and only if there exists a tree $T$ with its maximum degree $\mathrm{\Delta }(T)\le r$ and Laplacian spectral radius $\mu (T)=\alpha >1$. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers $\alpha \ge 2$ such that $(\alpha -1{)}^{-1}\in {\mathcal{L}}_r(\alpha )$ for $r=\lfloor \alpha \rfloor -1$. Applying this conclusion, we then show that for any integer $n$, there exists a tree $T$ with $\mathrm{\Delta }(T)<n$ and $\mu (T)=n+1$ if and only if $n\ge 4$.