  We present a detailed study of the star formation history (SFH) of the Tucana
dwarf spheroidal galaxy. High quality, deep HST/ACS data, allowed us to obtain
the deepest color-magnitude diagram to date, reaching the old main sequence
turnoff (F814 ~ 29) with good photometric accuracy. Our analysis, based on
three different SFH codes, shows that Tucana is an old and metal-poor stellar
system, which experienced a strong initial burst of star formation at a very
early epoch (~ 13 Gyr ago) which lasted a maximum of 1 Gyr (sigma value). We
are not able to unambiguously answer the question of whether most star
formation in Tucana occurred before or after the end of the reionization era,
and we analyze alternative scenarios that may explain the transformation of
Tucana from a gas-rich galaxy into a dSph. Current measurements of its radial
velocity do not preclude that Tucana may have crossed the inner regions of the
Local Group once, and so gas stripping by ram pressure and tides due to a close
interaction cannot be ruled out. On the other hand, the high star formation
rate measured at early times may have injected enough energy into the
interstellar medium to blow out a significant fraction of the initial gas
content. Gas that is heated but not blown out would also be more easily
stripped via ram pressure. We compare the SFH inferred for Tucana with that of
Cetus, the other isolated LG dSph galaxy in the LCID sample. We show that the
formation time of the bulk of star formation in Cetus is clearly delayed with
respect to that of Tucana. This reinforces the conclusion of Monelli et al.
(2010) that Cetus formed the vast majority of its stars after the end of the
reionization era implying, therefore, that small dwarf galaxies are not
necessarily strongly affected by reionization, in agreement with many
state-of-the-art cosmological models. [abridged]
