A. Before planting under a tree with lots of exposed roots, you must first build-up the soil for the new plants to grow. At the same time, the tree needs the oxygen that permeates the soil at the surface level in order to carry out the various chemical processes that take place in plants to produce food. If too much soil is applied at any one time, the roots are likely to be deprived of oxygen and the tree will suffer.
I recommend 'pocket planting.' This simply means that you gently dig small holes big enough to plant a small plant in the spaces between the major roots. Choose shade tolerant plants, either ground covers or young perennials. Small transplants do not need to be very deep, so there should be minimal damage to the tree's root system.
Once everything is planted, mulch the area with a combination of well-rotted manure, compost and/or leaf mold to a depth of 5 cm. The tree roots will still be able to breathe. Keep the whole area moist for two to three weeks until your plants are established. In the first year, it may be necessary to water during a prolonged drought. You can apply the same type of mulch annually, but never more than a 5 cm layer in any one year.
A few examples of some shade-loving perennials: hostas, astilbe, tiarella
Some shade loving annuals: begonias, coleus, impatiens, lobelia
Some shade loving ground covers: Saxifragas x urbium 'Aureopunctata'
It is very difficult to grow large shrubs in a shade area with exposed roots.