We detail the rich electronic and vibrational structure of triatomic 'butterfly' molecules, ultra-long-range Rydberg molecules bound by resonant p-wave scattering. We divide these molecules into two sub-classes depending on their parity under reflection of the electronic wave function through the molecular plane. The trimers with odd reflection parity have smoothly varying, non-oscillatory potential energy surfaces except near the collinear configuration. Here, the vibrational wave function is confined tightly in the symmetric-stretch and bending modes, but only loosely in the asymmetric stretch mode. The trimers with even reflection parity exhibit far richer potential surfaces with abundant minima, but only a few of these are deep enough to localize the vibrational states. These minima are correlated with the electronic wave functions of the butterfly dimer, contributing to a building principle for trimers.