I grabbed a bunch of cats from the gallery and had them all use the cat tree with the selectable pets mod. They kept running off while I was controlling the other cats so I couldn't get a screenshot showing cats on every level at once... But I was able to test every slot (highest level, second-highest level, cube, bottom level, and the hammock. It seems the whole tree does function.
edit to add screenshots using the big tree (not the separate medium tree off to the side):
https://live.staticflickr.com/65535/50203962053_88ba0fd6e1_b.jpg
https://live.staticflickr.com/65535/50204767647_0d15619002_b.jpg