  The titanium abundances of late-type stars are important tracers of Galactic
formation history. However, abundances inferred from Ti I and Ti II lines can
be in stark disagreement in very metal-poor giants. Departures from local
thermodynamic equilibrium (LTE) have a large impact on the minority neutral
species and thus influences the ionisation imbalance, but satisfactory non-LTE
modelling for both dwarfs and giants has not been achieved in previous
literature. The reliability of titanium abundances is reassessed in benchmark
dwarfs and giants using a new non-LTE model and one-dimensional (1D) model
atmospheres. A comprehensive model atom was compiled with a more extended level
structure and newly published data for inelastic collisions between Ti I and
neutral hydrogen. In 1D LTE, the Ti I and Ti II lines agree to within $0.06$
dex for the Sun, Arcturus, and the very metal-poor stars HD84937 and HD140283.
For the very metal-poor giant HD122563, the Ti I lines give an abundance that
is $0.47$ dex lower than that from Ti II. The 1D non-LTE corrections can reach
$+0.4$ dex for individual Ti I lines and $+0.1$ dex for individual Ti II lines,
and reduce the overall ionisation imbalance to $-0.17$ dex for HD122563.
However, it also increases the imbalance for the very metal-poor dwarf and
sub-giant to around $0.2$ dex. Using 1D non-LTE reduces the ionisation
imbalance in very metal-poor giants but breaks the balance of other very
metal-poor stars, consistent with the conclusions in earlier literature. To
make further progress, consistent 3D non-LTE models are needed.
