The best option would be Elmer’s Wood Glue. This is a non-toxic glue, easy to clean up, and cheaper than buying a new tree! Even though it’s not toxic, you shouldn’t let your cat ingest it, since it can cause distress in her GI tract.
Just apply the glue to the cat tree, and then add the sisal rope or carpet. If you’re doing minor repairs, it’s not necessary to use hot glue. Once you’ve applied the glue, you’ll need to press down on the area for a few minutes.
The carpet on cat trees usually get loose, and wood glue would be enough to repair this. But, if you need to replace a bigger area, you can pick up carpet scraps from different places – such as a big store that’s going to throw them anyway.