  We discuss flavor-mixing probabilities and flavor ratios of high energy
astrophysical neutrinos. In the first part of this paper, we expand the
neutrino flavor-fluxes in terms of the small parameters U_{e3} and pi/4 -
theta_{23}, and show that there are universal first and second order
corrections. The second order term can exceed the first order term, and so
should be included in any analytic study. We also investigate the probabilities
and ratios after a further expansion around the tribimaximal value of sin^2
theta_{12} = 1/3. In the second part of the paper, we discuss implications of
deviations of initial flavor ratios from the usually assumed, idealized flavor
compositions for pion, muon-damped, and neutron beam sources, viz., (1 : 2 :
0), (0 : 1 : 0), and (1 : 0 : 0), respectively. We show that even small
deviations have significant consequences for the observed flavor ratios at
Earth. If initial flavor deviations are not taken into account in analyses,
then false inferences for the values in the PMNS matrix elements (angles and
phase) may result.
