A diffuse flux of astrophysical neutrinos above 100 TeV has been observed at the IceCube Neutrino Observatory. Here we extend this analysis to probe the astrophysical flux down to 35 TeV and analyze its flavor composition by classifying events as showers or tracks. Taking advantage of lower atmospheric backgrounds for showerlike 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))(circle plus) approximate to (1 : 1 : 1)(circle plus) flavor ratio at Earth commonly expected from the averaged oscillations of neutrinos produced by pion decay in distant astrophysical sources. Limits are placed on nonstandard flavor compositions that cannot be produced by averaged neutrino oscillations but could arise in exotic physics scenarios. A maximally tracklike composition of (0 : 1 : 0)(circle plus) is excluded at 3.3 sigma, and a purely showerlike composition of (1 : 0 : 0)(circle plus) is excluded at 2.3 sigma.