We had a storm last night, and some small branches from a next-door tree were knocked down in my yard. These branches have leaves with odd nodules at the base of many of them. See picture. What is this tree, and what are these nodules?

Ah, the nodules, when you break them open, are filled with tiny white wiggly things clinging to the inside of the cavity inside. So the nodules are caused by some invasive insect.