Due to the crowded nature of our "fence row" of trees, none of their branches are really suitable for swings. But they're our only mature trees. It's going to be a while before the trees we planted after moving in will work for this, but there are some promising-looking branches. I am eyeing Jim's mimosa tree, which could probably support a baby swing at any rate. :-)
I've also got this idea about painting a tree in the corner of Sunshine Girl's room, extending the branches out along the ceiling, and hanging a swing for her... a baby swing for now, later a playground style swing, and when she's a teen, replace it with a hammock chair. DH says "maybe" but wants to anchor it through a ceiling stud, with eyebolts as you suggested doing with a tree swing.
Our first swing was a baby one and hung from a crabapple branch about two inches diameter. With the small load and short straps it could hang fairly close to the trunk. I was fine, no sway or bounce that looked like we overloaded the branch. This rnangy mostly unpruned crab has now had swings in a couple other places.
Great fun idea for the kid's room! I got a hammock chair for outside and its very comfortable.
Thanks for reading and commenting ^_^