  We study limit points of the spectral radii of Laplacian matrices of graphs.
We adapted the method used by J. B. Shearer in 1989, devised to prove the
density of adjacency limit points of caterpillars, to Laplacian limit points.
We show that this fails, in the sense that there is an interval for which the
method produces no limit points. Then we generalize the method to Laplacian
limit points of linear trees and prove that it generates a larger set of limit
points. The results of this manuscript may provide important tools for proving
the density of Laplacian limit points in $[4.38+, \infty)$.
