Answer: Planting under a tree is not too different from planting anywhere else, but to some extent your options will be dictated by the type of tree it is.
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 for a beginner would 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 type of tree and which side of it you are planting on) so you will have fun experimenting.
When preparing the soil, work around the large roots to avoid injuring them, and do not change the soil level substantially. Some trees, such as oaks and firs, are very sensitive to changes in grade.
Some trees, including dogwoods, are very sensitive to any root disturbance, but others are more resiliant. Planting under a tree with numerous surface roots as one finds, for example, with some types of maples can be very difficult because the tree roots take up nearly all the soil moisture and nutrients. In such a situation, it may be better to select a different location for your garden or possibly use an aggressive evergreen groundcover such as English ivy (Hedera helix) or Periwinkle (Vinca minor) along with several large containers filled with shade tolerant annuals for extra seasonal color. To establish the groundcover, allow between one plant per square foot or one per two square feet depending on how fast you would like the area covered. You will need to dig an individual planting hole for each small starter plant, removing a section of the tree roots in order to give the plant a root-free area in which to take hold, improve the soil by adding organic matter such as compost and then plant.
Whichever appraoch you take, 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 down weeds until the ground cover fills in. If weeds are a problem in that area, you may also use several thicknesses of wet newspaper beneath the mulch to help suppress them.
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