Living with cats and their problem of not being able to spit anything but pills out of their mouths, let me gently suggest that you keep an eye on what goes into and comes out of that Great Dane.
If the dog ate what seems like a significant volume of the tree skirt, you might consider giving your vet a call and asking what to do to "help things along." I can tell you that cats CAN safely take stool softeners. But, that's cats and your vet would absolutely be the place to ask technical questions. Mine has been together with us so long he doesn't turn a hair at off the wall questions on the phone.
Since I do not want to risk offending, let me wish you a Happy New Year a bit early.
Pat