I have repaired parts of a cat tree. Somewhere buried in that carpet is the head of a screw. Find the screws and remove the shelf rrom the tree, since that will be the easiest way to cover it. Buy the type carpet you want and then use glue, small tacks and screws to hold it in place as you mold it to the shape of the shelves....using the majority of the tacks and screws on the undersides. The carpet with hide the nail/screw heads.

Use area rug rather than carpet, it's thinner and easier to work with. Jute backed carpet is very stiff. Don't get looped carpet rugs, cats claws get caught. The best luck I've have finding area rugs, with rubber backing (thin enough to go in washing machine) the kind you wash is at Target. I use a electric stapler or U shaped tacks you pound in with hammer.....or get some remenent carpet if you want to try regular carpet, the thinest you can get. But nothing with loops.

Thank you both. I will make this my new project. The tree has been well loved like the velvetine rabbit since 1989 :)