We constrain the assembly history of galaxies in the projected central 0.5 Mpc of the Coma cluster by performing structural decomposition on 69 massive (M⋆ ≥ 109 M⊙) galaxies using high-resolution F814W images from the Hubble Space Telescope (HST) Treasury Survey of Coma. Each galaxy is modelled with up to three Sérsic components having a free Sérsic index n. After excluding the two cDs in the projected central 0.5 Mpc of Coma, 57 per cent of the galactic stellar mass in the projected central 0.5 Mpc of Coma resides in classical bulges/ellipticals while 43 per cent resides in cold disc-dominated structures. Most of the stellar mass in Coma may have been assembled through major (and possibly minor) mergers. Hubble types are assigned based on the decompositions, and we find a strong morphology-density relation; the ratio of (E+S0):spirals is (91.0 per cent):9.0 per cent. In agreement with earlier work, the size of outer discs in Coma S0s/spirals is smaller compared with lower density environments captured with SDSS (Data Release 2). Among similar-mass clusters from a hierarchical semi-analytic model, no single cluster can simultaneously match all the global properties of the Coma cluster. The model strongly overpredicts the mass of cold gas and underpredicts the mean fraction of stellar mass locked in hot components over a wide range of galaxy masses. We suggest that these disagreements with the model result from missing cluster physics (e.g. ram-pressure stripping), and certain bulge assembly modes (e.g. mergers of clumps). Overall, our study of Coma underscores that galaxy evolution is not solely a function of stellar mass, but also of environment.