I didn't really plan it, so sequencing building the frame vs attaching the carpet was a little annoying, but overall not bad at all.

This isn't the first one I've made for them, but I used spray adhesive on the first, which wasn't nearly effective enough. This time I used carpet adhesive, which was more cost effective and seems to be much more effective. Downside to that, though is that I had to wait 24 hours after carpeting a piece to attach it

I'd recommend using something like 3/8 ply, instead of the 3/4 I used, you'll save a ton of weight. For joining vertical and horizontal surfaces I used a few, small L-brackets and a couple blocks that doubled as reinforcement

For the record, if you go to carpet stores you can frequently get discounts by getting discards, or like, broken packs of the tiles.

The trouble is, the scraps aren’t always what you’re looking for.

I'm not OP, but I recently rewrapped my cat trees and used a staple gun to tack it down. Close to the base, I wrapped the tail vertically a little upwards on the pole, stapled it to secure, then ran it down to the base and started winding over the tail. This causes some bumps in the snaked look, but isn't too bad from a distance. Plenty of staples keep it in place up the pole in case the cats cut through it from sharpening.

I drilled two holes at the ends of the post. Tied a knot in the top end, wound my way down, then ended with a clove hitch to maintain three tension, and put the excess through the other hole and tied another knot just for extra security

I don't know about OP, but when I put rope on my cat-tree's soft-fabric column I just wound it around the column (cylinder, don't know if it would work as well on a squarer support) as tightly as I could and safety pinned the bottom end to the 2nd to last row. The tight coil encourages friction and prevents slippage when he scratches it. The rope hasn't moved, aside from when I rotate it for a fresher side, since I placed it over 2 years ago.

Given that both ends pass through a hole in the post, to the back side that we can't see, I'm going to guess that knots are holding it in place.