Hard for me to see how the area underneath a tree is a full sun area, but whatever. Check out the groundcover thymes offered by High Country Gardens.

It's a partially shaded area. It gets filtered sun.

Filtered sun entirely different. Look into Geranium macrorrhizum--Big root geranium--deer resistant, semi-evergreen in my zone, dense, easy to control, weed suppressing, and flowers in late spring. No attention needed.

A couple of nice groundcovers that could work would be plumbago and Veronica pectinata.

If by plumbago the previous poster means Ceratostigma plumbaginoides, yes, that would work. But unlike the G. macrorrhizum, it is not evergreen and requires cutting back or mowing in fall after freeze or in early spring.