The best groundcover to plant under a tree where nothing else grows is liriope(monkey grass).It is not harmful to animals and can even take their abuse(peeing,pooping,laying and even walking on it.Monkey grass is for zones 4-10.It is quite pricey where I live but it is on ebay and other sites.

I do not recommend english ivy to anyone.It will climb the tree and eventually it will cover the branches and then the tree cannot leaf out and will die.

Monkey grass grows anywhere in full shade or full sun.Hope this helps.

I will second this suggestion. The prior owners of our house used it as a ground cover under and around large oak trees in the back beside a high deck. This bed is shaded and the little ground cover does wonderfully well. We have a large dog and his activity doesn't damage the planting, plus I walk over it to reach a water faucet. It is very hardy and tough and stays green year round. If they wish to add a bit of color, just place three pots of varying heights together and put some bright Impatients in them. Lovely and cool.

Mary

Here is a link that might be useful: Mondo Grass photo and description