  We study the eigenvectors of Laplacian matrices of trees. The Laplacian
matrix is reduced to a tridiagonal matrix using the Schur complement. This
preserves the eigenvectors and allows us to provide fomulas for the ratio of
eigenvector entries. We also obtain bounds on the ratio of eigenvector entries
along a path in terms of the eigenvalue and Perron values. The results are then
applied to the Fiedler vector. Here we locate the extremal entries of the
Fiedler vector and study classes of graphs such that the extremal entries can
be found at the end points of the longest path.
