Fastest growing (your lifetime!) here in Maine are spruce, honey locust, silver maple, chinese elm, and wild black cherry. Much depends on your soil and how cold it gets. Our trees have to survive zone 3 if you want them around for a lifetime or two.
My favorite shade trees were the sugar maples we had as a kid, but then they were well over a 100 years old!
- as for invasive - each state has their forbidden list. Outside of that list, I wouldn't worry about it.
- Once found a red rosa rugosa growing on the midst of our woods. Obviously an invasive plant there by its lonesome in the shade.
I dug it up and moved it to the front yard!
Pesky birds. Just like with all of the apple trees they plant and the horses harvest. Pesky invasives!