back up this info with your own research because I'm new and don;t want to lead you astray but also been looking into this:
you want static rope, not dynamic, and never ever ever climb above your tether. get a climbing rated rope NOT one from a big box store/ hardware store. check out doublesteps.com, lots of people like that site and the guy that runs it is a member here
@DanO. sterling oplux is 8mm (doublesteps also carries another 8mm rope that's pretty much identical) is about the thinnest you'll want but be warned that diameter isn't rated for use with a LOT of mechanical devices/ATCs so you may want to stick with a thicker rope. thicker rope = more friction in the same device= safer rappel (in a simplistic newbie way, this is MUCH more nuanced and rope/device rope/prussik combos play a big factor too). many people use ropes from 9-11mm and they work great, just slightly bulkier/heavier to pack in and out.
a lot of people will use devices that are slightly out of spec- the madrock safeguard is a popular rappeling option, and sometimes is used on 8mm rope though i think it's only rated down to 8.1? There's also a large crew that prefers an ATC or tubular belay device backed up with an autoblock/prussik (or similar style) knot on the free "tag end" side some also use a blake's hitch or other arborist knots. some people also use a prussik higher up on the tree side above your rapelling device as a redundant safety.
there's an overwhelming amount of into to research, it'll all start making sense soon, hope that helps you get started. (and again, please verify anything i've said elsewhere!)