I had the same situation with large fir trees with shallow roots surrounding the run and coop. I was afraid that flooring the run with wire would be hard on the chickens' feet when they scratched and dustbathed in the soft, but very rooty and dry ground under the firs.
I actually bought welded wire (3 feet wide with 1" x 2" ) and ran it horizontally from the base of the coop and run outwards, like a Christmas tree skirt without digging it in. I used baling wire and/or heavy duty zip ties every couple of inches to connect the skirt to the hardware cloth that is the fencing of the run. I do not have to worry about weasels or other very small predators, but I have a dachshund who digs and possums and raccoons around regularly.
The grass and groundcover are growing right up through the wire, which was secured with a few landscape tacks to keep it flat on the ground as the plants grew in. It's pretty invisible now, but if any critter tries to dig in at the perimeter, they encounter the welded wire and cannot get in. I have found evidence of attempts to dig, but no intruders have made it into the run.
I also used welded wire as a roof for the run area; although the doxie can't climb, there are many overhanging trees and I didn't want a cat or coon dropping into the run from the trees or the roof of the coop itself. As far as whether coons will dig their way in... I just assume that they will, although I have no direct knowledge. They are persistent and pretty smart. Possums will dig.
Just about every part of my run and coop are found/recycled, but I did invest in many, many feet of hardware cloth and welded wire to predator proof. And remember... chicken wire is designed to keep chickens in, not predators out. It's too flimsy for predator proofing and the coons will chew right through it to get what they want. And if they don't chew through, they'll reach in and take handfuls of chicken and kill them that way. 1/2' x 1/2" hardware (or "masons'") cloth is the best thing to use for at least the bottom few feet of your fencing, as the coons can't chew or reach through.
