They really like staying in protected areas. I think their instincts are more that they like protection from flying things like hawks than ground predators. They love scratching though leaves and stuff like that. They use the shade in the heat of summer to stay cool. The ideal set-up for you would be to have a bit of both woods and open area in both runs if you can swing that. They like grazing in the vegetation in the open areas too. I’ll bet they’ll make their dust baths at the verge between the woods and open area.
One thing to consider in building a run in the woods is that some predators can cross between the treetops and climb down into your run. Most of these predators are going to be more active at night, but not always. I don’t want to scare you away from doing this because of something that might, possibly, occasionally, sometimes, could happen but your risks do go up. I’d suggest you be really diligent in locking them in a safe coop at night. That’s when your risks really go up, nighttime.