  The paper studies the spectral properties of the Schr\"odinger operator
$A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying
real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the
free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single
eigenvalue of infinite multiplicity in the middle of each finite gap. The
perturbation $gV$ gives rise to extra eigenvalues in the gaps. These
eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed
sign. Assuming that the latter condition is satisfied and that $V$ is
symmetric, i.e. depends on the distance to the root of the tree, we carry out a
detailed asymptotic analysis of the counting function of the discrete
eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$,
this asymptotics is either of the Weyl type or is completely determined by the
behaviour of $V$ at infinity.
