Answer: Many groundcovers can be divided, but it would depend on what kind you have as to how and when is best to do it. Some such as vinca minor might be more easily increased by rooting tip cuttings or by buying the inexpensive small plants that are sold in the spring for this purpose. Others such as spreading junipers would not divide successfully. Your local county extension and/or store staff should be able to help you identify which plant you have and determine which way would be best to increase it.
As far as the tree goes, most deciduous trees do lose all their leaves in the fall, but a few may retain some through the next season. Since you described them as reddish, I suspect it may be one of the varieties of oak that routinely keep some of their foliage each winter. Again, someone locally should be able to help you identify it based on a photo or a sample.
I'm sorry I can't be more specific for you but I hope this helps you get started.
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