Answer: Planting under a tree is not too different from planting anywhere else.
The soil needs to be loosened, you need ot add organic matter to it, and the plants need to be shade tolerant. Some easy but rewarding plants include hosta (there are many different varieties of hosta available with foliage that is green, blue, golden, striped, textured, and so on), tricyrtis, Solomon's Seal, epimedium, lamium and pulmonaria (lungwort) for deep to moderate shade; and astilbe, bleeding heart and perennial geranium for light shade. There are many more plants that will grow well in partial shade (shade can vary with the size of tree.
When preparing the soil, work around the large roots to avoid injuring them, and do not change the soil level substantially. Some trees are very sensitive to any root disturbance, but others are more resiliant.
Be sure to water well and keep the soil evenly moist but not soggy until the new plants become established. Between the little plants you can mulch with several inches of organic mulch to keep weeds down.
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