  We have conducted a near-infrared monitoring campaign at the UK InfraRed
Telescope (UKIRT), of the Local Group spiral galaxy M33 (Triangulum). The main
aim was to identify stars in the very final stage of their evolution, and for
which the luminosity is more directly related to the birth mass than the more
numerous less-evolved giant stars that continue to increase in luminosity. In
this second paper of the series, we construct the birth mass function and hence
derive the star formation history. The star formation rate has varied between
~0.002 and 0.007 M$_\odot$ yr^-1 kpc^-2. We give evidence of two epochs of a
star formation rate enhanced by a factor of a few -- one that happened \geq 6
Gyr ago and produced \geq 80% of the total mass in stars, and one around 250
Myr ago that lasted ~ 200 Myr and formed \leq 6% of the mass in stars. We
construct radial and azimuthal distributions in the image plane and in the
galaxy plane for populations associated with old first-ascent red giant branch
(RGB) stars, intermediate-age Asymptotic Giant Branch (AGB) stars and young
(massive) blue and red supergiants. We find that the RGB stars follow a
spheroidal distribution, while younger stars follow a flat-disc distribution.
The intermediate-age population displays signs of a pseudo-bulge or possibly a
bar. The inner spiral arm pattern as recorded in mid-19^th-century drawings is
confirmed. We interpret our findings as evidence for an old, pressure-supported
component and a younger disc formed 6 Gyr ago, with an accretion event
occurring 250 Myr ago giving rise to the compact nucleus in M33. Our study
provides support for recent Padova stellar evolution models except that
super-AGB stars likely reach low temperatures and thus high mass-loss rates,
supporting the super-AGB nature of the progenitors of dust-enshrouded
supernovae such as SN2008S.
