  We discuss the transition from a fully decoherent to a (quasi-)condensate
regime in a harmonically trapped weakly interacting 1D Bose gas. By using
analytic approaches and verifying them against exact numerical solutions, we
find a characteristic crossover temperature and crossover atom number that
depend on the interaction strength and the trap frequency. We then identify the
conditions for observing either an interaction-induced crossover scenario or
else a finite-size Bose-Einstein condensation phenomenon characteristic of an
\textit{ideal} trapped 1D gas.
