  We study fluctuations of the local density of states (LDOS) on a tree-like
lattice with large branching number $m$. The average form of the local spectral
function (at given value of the random potential in the observation point)
shows a crossover from the Lorentzian to semicircular form at $\alpha\sim 1/m$,
where $\alpha= (V/W)^2$, $V$ is the typical value of the hopping matrix
element, and $W$ is the width of the distribution of random site energies. For
$\alpha>1/m^2$ the LDOS fluctuations (with respect to this average form) are
weak. In the opposite case, $\alpha<1/m^2$, the fluctuations get strong and the
average LDOS ceases to be representative, which is related to the existence of
the Anderson transition at $\alpha_c\sim 1/(m^2\log^2m)$. On the localized side
of the transition the spectrum is discrete, and LDOS is given by a set of
$\delta$-like peaks. The effective number of components in this regime is given
by $1/P$, with $P$ being the inverse participation ratio. It is shown that $P$
has in the transition point a limiting value $P_c$ close to unity, $1-P_c\sim
1/\log m$, so that the system undergoes a transition directly from the deeply
localized to extended phase. On the side of delocalized states, the peaks in
LDOS get broadened, with a width $\sim\exp\{-{const}\log
m[(\alpha-\alpha_c)/\alpha_c]^{-1/2}\}$ being exponentially small near the
transition point. We discuss application of our results to the problem of the
quasiparticle line shape in a finite Fermi system, as suggested recently by
Altshuler, Gefen, Kamenev, and Levitov.
