  We study the internal structural properties of 37 long duration gamma-ray
burst (GRB) host galaxies imaged with the Hubble Space Telescope. Fitting
exponential disk and r^1/4 models to the surface brightness profiles of eight z
< 1.2 bright host galaxies, we find that the disk model is slightly preferred
for most hosts, although two galaxies are fit best with an r^1/4 profile. We
furthermore measure the central concentrations and asymmetries of all 37 host
galaxies using the CAS system, and compare with field galaxies in the Hubble
Deep Field. Our first main conclusion is that GRB hosts exhibit a surprisingly
high diversity of galaxy types. A significant fraction (68%) of host galaxies
are situated in a region of the concentration-asymmetry diagram occupied by
spirals or peculiar/merging galaxies. Twelve hosts (32%) are situated in the
region occupied by elliptical galaxies, having high concentration indices. We
also find some evidence for evolution in GRB host galaxy morphology, such that
hosts at z > 1 have a relatively high light concentration, indicating that
these systems are perhaps progenitors of massive galaxies, or are compact blue
star forming galaxies. We find that GRB hosts at z > 1 are different from the
general field population at z > 1 in terms of light concentration at >99.5%
confidence, yet have sizes similar to the general z > 1 population. This is the
opposite of the effect at z < 1 where GRB hosts are smaller than average. We
argue that GRB hosts trace the starburst population at high redshift, as
similarly concentrated galaxies at z > 1 are undergoing a disproportionate
amount of star formation for their luminosities. Our results show that GRBs are
perhaps ideal tracers of typical galaxies undergoing star formation at any
epoch.
