  We present a statistical characterization of the carbon-star to M-giant (C/M)
ratio in the halo of M31. Based on application of pseudo-filter band passes to
our Keck/DEIMOS spectra we measure the 81-77-color index of 1288 stars in the
giant stellar stream and in halo fields out to large distances. From this
well-established narrow-band system, supplemented by V-I colors, we find only a
low number (five in total) of C-star candidates. The resulting low C/M ratio of
10% is consistent with the values in the M31 disk and inner halo from the
literature. Although our analysis is challenged by small number statistics and
our sample selection, there is an indication that the oxygen-rich M-giants
occur in similar number throughout the entire halo. We also find no difference
in the C-star population of the halo fields compared to the giant stream. The
very low C/M ratio is at odds with the observed low metallicities and the
presence of intermediate-age stars at large radii. Our observed absence of a
substantial carbon star population in the these regions indicates that the
(outer) M31 halo cannot be dominated by the debris of disk-like or SMC-type
galaxies, but rather resemble the dwarf elliptical NGC 147.
