  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 topographically
smooth 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.
