  Given a smooth target curve $X$, we explore the relationship between
Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold
Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved
that relative invariants are equal to the $r^0$-coefficient of the
corresponding orbifold Gromov-Witten invariants of $r$-th root stack for $r$
sufficiently large. Our result provides a precise relation between relative and
orbifold invariants of target curves generalizing the result of
Abramovich-Cadman-Wise to higher genus invariants of curves. Moreover, when $r$
is sufficiently large, we proved that relative stationary invariants of $X$ are
equal to the orbifold stationary invariants in all genera.
  Our results lead to some interesting applications: a new proof of genus zero
equality between relative and orbifold invariants of $X$ via localization; a
new proof of the formula of Johnson-Pandharipande-Tseng for double Hurwitz
numbers; a version of GW/H correspondence for stationary orbifold invariants.
