Too bad about your tree.
I have built many treehouses, and one of the simplest for small children is a platform that is only partially supported by the tree. I put two 2x6's, one on either side of the tree, with the far ends supported by posts. Much like the "H" described by a previous poster, but off-center. Then, on top of that "H", you frame up whatever you want. In general, simpler is better, kinda like your kids getting lots of cool gifts at Christmas, but only wanting to play with the refrigerator box. My kids played for years in a little platform treehouse with low (24") walls, and no roof. It had a zipline to a neighboring tree, which also functioned well as a rope swing. Regarding ziplines, if you can get steel cable, it really improves the performance and reliability. I used a used 7/32" cable (free) and my neighbors' linebacker-sized friends came over and played on it, never even stretched. With rope, you will never get it tight enough, and the incline will have to be significant.
Whatever you do, with your trees, just getting your kid outside, away from the computer & TV is an accomplishment.
Welcome to A2K! Lots of great info there, thanks. I did in fact make the rope swing and my kid (and her friends!) have been having a ton of fun with it. I still like the idea of a treehouse and may well come back to that idea later.