We present detailed NLTE Ti I calculations in model atmospheres of coo
l dwarf and giant stars. A fully self-consistent NLTE treatment for a
Ti I model atom with 395 levels and 5279 primary bound-bound transitio
ns is included, and we discuss the implication of departures from LTE
in this atom for the strengths of Ti I lines and TiO molecular bands i
n cool star spectra. We show that in the atmospheric parameter range i
nvestigated, LTE is a poor approximation to Ti I line formation, as ex
pected from the low collisional rates in cool stars. The secondary eff
ects of Ti I overionization on the TiO number density and the TiO mole
cular opacities, however, are found to be negligible in the molecular
line-forming region for the relatively small parameter range studied i
n this paper.