  A diffuse flux of astrophysical neutrinos above $100\,\mathrm{TeV}$ has been
observed at the IceCube Neutrino Observatory. Here we extend this analysis to
probe the astrophysical flux down to $35\,\mathrm{TeV}$ and analyze its flavor
composition by classifying events as showers or tracks. Taking advantage of
lower atmospheric backgrounds for shower-like events, we obtain a shower-biased
sample containing 129 showers and 8 tracks collected in three years from 2010
to 2013. We demonstrate consistency with the
$(f_e:f_{\mu}:f_\tau)_\oplus\approx(1:1:1)_\oplus$ flavor ratio at Earth
commonly expected from the averaged oscillations of neutrinos produced by pion
decay in distant astrophysical sources. Limits are placed on non-standard
flavor compositions that cannot be produced by averaged neutrino oscillations
but could arise in exotic physics scenarios. A maximally track-like composition
of $(0:1:0)_\oplus$ is excluded at $3.3\sigma$, and a purely shower-like
composition of $(1:0:0)_\oplus$ is excluded at $2.3\sigma$.
