Answer: The dappled light under tall oak trees is often suitable for azaleas and rhododendrons, and since these are oaks the soil should be somewhat acid which is perfect for those shrubs. You could also try yews and hollies since they tolerate some shade, but these are naturally far larger than three to five feet tall and would require pruning. Perennials to consider would be hosta, pulmonaria, bleeding heart, ferns, Solomon's seal, tricyrtis and ajuga.
The moss may be an indicator of compacted soil, extremely acid soil, infertile soil or poorly drained soil or any combination of those. Knowing that, no matter what you plant, you will need to pay some attention to soil preparation first.
Unfortunately, deer will often feast on all of the above shrubs and adore hosta. Additional (deciduous) shrubs to consider with slightly less attraction to deer are itea and clethra. However, deer will eat nearly anything if they are hungry or if the local population has realized it is edible. You might want to check with your neighbors to see which plants they are still able to grow without deer damage; you might also consider a fence to protect your plantings because unfortunately, that is the most effective long term solution to that problem.
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