  We study the discreteness of the spectrum of Schrodinger operators which are
defined on N-dimensional rooted trees of a finite or infinite volume, and are
subject to a certain mixed boundary condition. We present a method to estimate
their eigenvalues using operators on a one-dimensional tree. These operators
are called width-weighted operators, since their coefficients depend on the
section width or area of the N-dimensional tree. We show that the spectrum of
the width-weighted operator tends to the spectrum of a one-dimensional limit
operator as the sections width tends to zero. Moreover, the projections to the
one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator
converge to the corresponding eigenfunctions of the one-dimensional limit
operator.
