Purple Hearts (Tradescantia pallida 'Purple Heart') has a very fast sprawling growth rate and is great for flower beds and groundcover! I have this in my flower beds and it is so pretty, grows fast and roots easily!
Epimedium Red Barrenwort -Berberidaceae can tolerate dry shade such as the base of a tree(once established). Red spring flowers, heart shaped leaves, and bronze fall foliage make this a multi-functional plant.
Kennedia prostrata-Kennedia prostrate: Low-growing plant suitable as ground cover or hanging plant. Propagation is easy from seed. The plant produces showy frequent red flowers when grown under bright sunlight.