Basically, you can't, especially if the trees aren't yours. It's very hard to dig and plant in such conditions. Are the trees also giving shade to your garden? Are they deciduous? I think one thing you could do is create a woodland garden - buy English bluebells in the green in spring, snowdrops in the green, Cyclamen hederifolium & C. coum. In the meantime, try to plant bulbs in between the roots, like crocuses, scillas, chionodoxa and Anemone blanda - all en masse.

Other small woodland plants that would thive are Omphalodes verna, O. cappadocica, Geranium maccrorrhizum, G. nodusum and G. versicolor, Gallium, Ajuga. All these will spread through and over the roots once they get going.

Shallow rooted plants will probably survive. Vinca does well - the variegated sorts lighten up darker areas and don't spread quite as much as the all green ones, though it is still a bit of a thug.

you could always put pots and shade lovers around the base or put some decking over them and use different size pots for interest .

I would line the bottom and sides of the raised beds with weed barrier to keep tree roots from creeping in, too.

good thinking tugbreth