You can bore holes in the tree stump and pour stump killer in them, which will obviously take time, or get a tree surgeon round with a stump grinder.

Plants under a tree - is it evergreen or deciduous? Under my weeping ash, I grow snowdrops and Cyclamen coum, which flower before the leaves start to appear. An evergreen would be much more difficult.

have a look at what i grow under a large bech tree.