Planting under mature maple trees poses two challenges: the deep shade that maples typically cast, and their thick, abundant roots. Of these two, the shade is definitely the easier one to deal with, not least of all because the roots pose two problems: one, the difficult of planting when the ground is mostly taken up by thick, woody roots, and two, the competitive advantage for water uptake that the established trees enjoy. This means you need to pick plants that can withstand some dry conditions, and that are available in small sizes so they are easy to plant. And you have to be prepared to provide extra water to help the plants get established while the trees attempt to suck it all up. Our list of suggestions that should be able to withstand these conditions include ajuga, perennial geranium, carex, brunnera, wintercreeper, ferns, and Japanese forest grass; we suggest you avoid myrtle/vinca/periwinkle and ivy. People may tell you these plants are tough as nails, which is true, but they are also invasive and difficult to manage so while planting them may solve your under-tree issue, it’s likely to cause new management issues. They’re best avoided. Finally, whatever you decide to plant, we suggest planting as soon as possible in spring. You want those new plants to be well-established before fall when the maples drop their leaves so that as you rake, you won’t be uprooting all the hard work planting work you did.