Creeping jenny sucks up water like a sponge and can turn brown if it does not get enough of it. It grows in water as an aquatic plant here if CA. I live in the woods and the best ground cover I've had is archangel lamium which is what the photo is of. It does well under trees and can grow in dark shade. The deer hate it because its aromatic and snails won't touch it. The roots are matting surface roots that can smother other weeds. You may need to clear it a bit around your host as to not let it overtake them. It stays fairly low but can send out some long twining stems. I have it mixed with other ground covers that it works well with. Baby tears around your tree trunks might be a nice accent if that works in zone 7. Archangel lamium gets yellow flowers on it in the spring.

Yellow Archangel is very invasive here in the Seattle area. Sunset says it's easily curbed, but I haven't found that to be true in my woods where it reseeds and has been VERY difficult to eradicate.
I've been using vinca minor and hardy geranium to keep most weeds under control in a woodsy environment of trees and shrubs.
Mike
Hardy geranium.

Here is a link that might be useful: My garden pics