Answer: Well, you have a lot of options we just need to narrow it down a bit. It is difficult to recommend plants without knowing a little more about your growing conditions. Consider these questions to help you clarify your needs.
Are you looking for zero maintenance on the slope? i.e., a ground cover that you don't have to mow at all? If yes on the ground cover, are you looking for flowers or foliage? Ajuga (bugleweed) is a durable, hardy ground cover that may fit the bill. You mention plants and bushes for that area...are you wanting to put a bunch of plants/bushes in and mulch the area to avoid mowing? Daylilies may be a good choice; they are hardy perennials that will spread over the years and require little maintenance.
How much flooding do you get (how many times a year, how much water, how long does it stand, etc.). Many plants cannot stand "wet feet", or saturated soil, so this is important to know.
Is the reason why the grass under the trees isn't growing because it's a deep shade area or because it stays wet? Both? What type of sun exposure does the slope and the area with trees get? These are also important considerations and will affect plant choices.
If we can eliminate some of these unknowns, I'm sure together we can figure out a plan of action. Feel free to submit another question (via the web site please) with more information, and we'll try to help!
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