  In this paper, we build up a min-max theory for minimal surfaces using
sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary
components. We show that the width for the area functional can be achieved by a
bubble tree limit consisting of branched genus $g$ free boundary minimal
surfaces with nodes, and possibly finitely many branched minimal spheres and
free boundary minimal disks.
