I have planted pears, arctic kiwi, and plums that seem to coexist fine with that weed. Your neighbors are more likely to be offended by it than your trees, but it won't help them much either. Dutch clover is better (very long nectar season, nitrogen fixing, leaves used by livestock and wildlife even though foreign, though produces similarly negligible biomass). However "creeping Charlie" does seem to grow through
wood chip mulches (which trees often like) very well. If you like Glechoma for tea or somesuch, you might be able to control it in a hanging basket. If you are thinking the smell of the foliage might deter pests that attack your trees, in my
experience the scent protects Glechoma alone.