Can bucket trucks be rented for trimming trees in my yard?

Depends on the area you live in. I wouldn't recommend it unless you have a professional tree-trimmer helping you. There are just too many dangers, plus some of them require 2 people anyway.

You are not likely to find anyone to rent and individual a bucket truck for personal use. You may, however, find someone with a bucket truck who it willing to trim your trees using it. There is too much liability for someone to rent you one.