Go to your local nursery and you can purchase plants that are specific for around the base of trees where it is almost impossible to grow grass. They can also be purchased online through nursery catalogs. They grow quickly and make for a beautiful landscape.
The easiest solution is to mulch around your trees instead of trying to get grass to grow. Some landscaping blocks and mulch make the muddy area look instantly beautiful and professional.
If you have done the standard things without any positive results, the best bet would be to consider using a particular grass, or grass blend, that has properties that grow well in shade.
