The Laplacian on a metric tree is $\Delta u = u''$ on its edges,
with the appropriate compatibility conditions at the vertices.
We study the eigenvalue problem on a rooted tree $\Gamma$:
$$-\lambda \Delta u = V u \quad \text{on } \Gamma,
\qquad u(o)=0.$$
Here $V \ge 0$ is a given ‘weight function’ on $\Gamma$,
and $o$ is the root of $\Gamma$.
The eigenvalues for such a problem decay no faster than
$\lambda_n = O(n^{-2})$, this last case being typical for
one-dimensional problems. We obtain estimates for the
eigenvalues in the classes $l_p$, with $p > \frac12$,
and their weak analogues $l_{p,\infty}$ with $p \ge \frac12$.
The results for $p<1$ and $p>1$ are of different character.
The case $p<1$ is studied in more detail. To analyse this
case, we use two methods; in both of them the problem is
reduced to a family of one-dimensional problems. One of the
methods is based upon a useful orthogonal decomposition
of the Dirichlet space on $\Gamma$. For a particular class of
trees and weights, this method leads to the complete
spectral analysis of the problem. We illustrate this in
several examples, where we were able to obtain the asymptotics
of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05;
secondary 34L15, 34L20.