Laplacian spectral radii α>1 of trees with max degree ≤r are exactly the numbers where (α-1)^{-1} lies in the recursively defined set L_r(α); this implies trees with Δ<n and μ=n+1 exist precisely for n≥4.
Stevanovi´ c, Bounding the largest eigenvalue of trees in terms of the largest vertex degree.Linear algebra and its applications360(2003), 35 – 42
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A new characterization of the set of Laplacian spectral radii of trees
Laplacian spectral radii α>1 of trees with max degree ≤r are exactly the numbers where (α-1)^{-1} lies in the recursively defined set L_r(α); this implies trees with Δ<n and μ=n+1 exist precisely for n≥4.