  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.
