I've done about 2 mi of dirt trail in groomed condition with a Scott SUB20 with the tire depicted below in 700x32. I didn't dare go fast since it's shared with pedestrians but the bike rode just fine.

I kind of assume you can see far enough ahead that you can avoid the roots though. If anything, I'd just throw some fatter tires on the bike to smooth out the ride, maybe 700x35 or 700x37. My daily commute bike has 700x42 and I can get it up to pretty good speed.