Large roots of mature trees are tricky to work around. In general we try to stay away from disturbing any major roots within the dripline of mature trees. Then it's about finding pockets where you can dig and not planting anything that's going to have a large woody root system of its own. You can see in this project, for example, that the area close to the trunk of that mature beech tree in the background is more or less bare, the rhododendron were there already, and the plants we added were groundcovers and perennials.

It also matters what the tree is. Norway Maples are notoriously difficult to plant under partially because of large aggressive root systems, but also because their canopies are so dense that very little light filters through to the ground beneath them. Black walnut trees actually produce an allelopathic chemical which impedes growth of other plants around its root system.