Just let them. They wasting their time and money if they do.

They’ll be able to sue you for branch removal costs. No problem — that shouldn’t cost more than a few thousand bucks.

Check your state's laws regarding trees. In most states, any branches hanging over a neighbor's yard is considered THEIR property, and therefore THEIR problem when they fall. Any part of a tree hanging over a property line isn't your tree parts.

Before it got cleaned up I hope you took lots of different angled photos...

Won't hold up in court, even if it had landed on one of them. Which is unfortunate it didn't.

Property law, they're entirely correct. And the reason they could sue is for endangerment. Even though it hit nothing on the way down, it was your responsibility to trim whatever hangs over into their property line. And since it could have caused bodily harm, and I'm willing to bet they've asked you to look at the tree previously too, no pity from me. Know your laws, be nice to your neighbors, and just adult the fuck up please? Can we all agree? This pity party needs to stop. Poor you? How about grow up YDI

If they want to sue you bring them to judge Judy. She will certainly see that what they’re saying is absolute and utter bullshit