We developed a procedure of reducing the number of vertices and edges of a given
tree, which we call the "tree simplification procedure," without changing its topological
information. Our motivation for developing this procedure was to reduce computational costs
of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of
trees representing dendritic structures of retinal ganglion cells of a mouse and computed
their graph Laplacian eigenvalues, we observed two "plateaux" (i.e., two sets of multiple
eigenvalues) in the eigenvalue distribution of each such simplified tree. In this article,
after describing our tree simplification procedure, we analyze why such eigenvalue plateaux
occur in a simplified tree, and explain such plateaux can occur in a more general graph if
it satisfies a certain condition, identify these two eigenvalues specifically as well as
the lower bound to their multiplicity.