We have performed a study of the one- and two-photon absorption properties of a systematically varied series of triarylamino-compounds with one, two, or three attached diarylborane arms arranged in linear dipolar, bent dipolar, and octupolar geometries. Two-photon fluorescence excitation spectra were measured over a wide spectral range with femtosecond laser pulses. We found that on going from the single-arm to the two- and three-arm systems, the peak in two-photon absorption (2PA) cross-section is suppressed by factors of 3-11 for the lowest excitonic level associated with the electronic coupling of the arms, whereas it is enhanced by factors of 4-8 for the higher excitonic level. These results show that the coupling of arms redistributes the 2PA cross-section between the excitonic levels in a manner that strongly favors the higher-energy excitonic level. The experimental data on one- and two-photon cross-sections, ground- and excited-state transition dipole moments, and permanent dipole moment differences between the ground and the lowest excited states were compared to the results obtained from a simple Frenkel exciton model and from highly correlated quantum-chemical calculations. It has been found that planarization of the structure around the triarylamine moiety leads to a sizable increase in peak 2PA cross-section for the lowest excitonic level of the two-arm system, whereas for the three-arm system, the corresponding peak was weakened and shifted to lower energy. Our studies show the importance of the interarm coupling, number of arms, and structural planarity on both the enhancement and the suppression of two-photon cross-sections in multiarm molecules.