Trees have very different soil needs than flowers and grasses. Thus the challenge is not just to re-engineer the clay soil in your area so that it is more flower and groundcover friendly, but also to alter the chemistry of the soil so that it accommodates the plants you want to grow without damaging the trees.
Trees use funghi to help them digest nutrients while grasses and flowers use bacteria. Tree soil conditions are the result of years of interactions. Clay is actually very rich in nutrients which is why so many forests have sadly been replaced by farmland. However, this change was achieved first at the expense of our forests areas and second through altering the soil so that it had a bacterial basis of nutrient absorption that crops (and flowers) prefer. Many trees fall due to human interventions in soil, particularly of the kind you are considering. See the Morton Arboretum white paper on soil considerations for trees here.
In order to minimize damage to the surrounding trees, I suggest that you first:
- Avoid bringing grasses and flower beds more than 1/4-1/3 of the way under the tree canopy as these areas are occupied by tree roots.
- You might even consider creating raised beds for flowers and production plants.
- Also, when altering clay soils you need to be careful. Many folks recommend adding sand, but this only works if you create a ratio of at least 1/3 to 1/2 sand to clay, otherwise you will end up with a concrete like mix if you use less sand.
- See Linda Chalker's Myth Buster white paper about adding sand to clay soil here.
Most people agree that mixing large amounts of sand and organic material like compost is the best way to amend clay and still benefits from the nutrients locked into the tighter particles that make up a clay soil. Alternately, adding organic material like composted leaves and mulch over a number of years will achieve the same end. There's more on that here.
Again the most cost effective method is to do this in raised beds with controlled amounts of material instead of massive re-engineering of the soil around your house which will inevitably negatively impact the surrounding trees and may run off over time depending on the slope of your property and what you plant.