  Using $J$ and $K_s$ band imaging obtained as part of the IMACS Cluster
Building Survey (ICBS) we measure S\'ersic indices for 2160 field and cluster
galaxies at $0.31 < z < 0.54$. Using both mass- and magnitude-limited samples,
we compare the distributions for spectroscopically determined passive,
continuously starforming, starburst, and poststarburst systems and show that
previously established spatial and statistical connections between these types
extend to their gross morphologies. Outside of cluster cores, we find close
structural ties between starburst and continuously starforming, as well as
poststarburst and passive types, but not between starbursts and poststarbursts.
These results independently support two conclusions presented in Paper II of
this ICBS series (Dressler et al.): 1) most starbursts are the product of a
non-disruptive triggering mechanism that is insensitive to global environment,
such as minor-mergers; 2) starbursts and poststarbursts generally represent
transient phases in the lives of "normal" starforming and quiescent galaxies,
respectively, originating from and returning to these systems in closed
"recycling" loops. In this picture, spectroscopically identified poststarbursts
constitute a minority of all recently terminated starbursts, largely ruling-out
the typical starburst as a quenching event in all but the densest environments.
