I have climbing roses growing through apple trees, and also honeysuckle.
That works, though you need to ensure that the particular climbing plant is not too vigourous for the tree. Clematis ought to work well too.
If attempting to grow low growing plants beneath a tree, then you'd need to bear in mind that the ground beneath a tree is likely to be shaded, and also relatively dry - so that's not such a good idea, IMO.