Answer: Unfortunately, it might not have been the pets' fault. Maple trees tend to have surface roots that make it difficult for other plants to grow beneath them becuase the roots steal all the moisture and nutrients and outcompete lesser plants. They also cast shade which means lawn grass is most likely not going to get as much sun as it would need to stay healthy.
Rather than try to grow lawn grass, you might want to consider using a shade tolerant groundcover that can grow in drier soil instead such as Vinca minor or Hosta lancifolia. These will need to be carefully watered until they become established, because of the tree roots. If you do not mind a natural look, violets and moss can be a lovely carpet beneath a tree as well.
If the tree is mature, it may be difficult to establish a planting area beneath it without damaging a large number of roots. This would not be good for the tree. Instead of planting the entire area, you might consider using a bark mulch under the tree. You could make it into a seating area, or add a birdbath or other focal point. For summer color you could grow annuals such as impatiens in a large container set beneath the tree.
Another option would be to encourage the moss as groundcover. This would be the least maintenance of all your options.
I hope this gives you some ideas.
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