Your cardboard beverage carton can be rolled into a rope sleeve. I'll share one method here that's been successful. By using a cellulose-based carton, you're drawing upon an abundant material to protect the tree. The leather sleeve is the standard for protecting the tree cambium from rope friction.
A leather sleeve will last many years - so I hear. I personally don't know. I seem to lose mine too often. At $10+ a sleeve, I didn't like how the cost was adding up. Thus, I began to make them out of beverage cartons. Since this was recycled fiber, my expectation was that they would last 6 - 12 climbs. Well, this type of cardboard sleeve is lasting beyond 20. The graphic side (colored outside) of the carton is slick and provides limited water protection. The inside (brown side) makes a good contact surface with glue.
Here's how to make one:
1. Roll the inner sleeve with graphics inside
2. Roll the outer sleeve with graphics out
3. Apply contact cement adhesive to the inside of #2 (brown side), and roll the two sleeves together (brown side glued to brown side)
4. Place on a flexible hose, and use electrical tape to bind the cartons together while the glue cures
5. Stuff inside a 5-gal pail
My scrap piece of hose had a ¾-inch inner diameter. I soon discovered on removing the cardboard sleeve from a tree that an in-line fig-8 could compress and slide through. Go figure! Thus select a hose with an inner diameter less than ¾-inch or be prepared to tie bulkier knots for a pull down.
At 60 F and 50% relative humidity, it takes up to a week for this to cure (cold basement conditions). Placing it in the hot sun would be preferred. When the solvent odor (MEK and toluene) dissipate, the glue is set. CAUTION: the odor of solvents signals that this material will ignite around an open flame. The solvents in the glue will also soften the carton. While it's soft, massage the curved shape. In the event that glue leaks on the rubber hose, slide it twice a day. By storing in a 5-gal pail, the shape is maintained until the glue stiffens it.
A greener approach would be to soak in water and just use tape. The contact cement method is more of a mass production approach and adds strength. To improve weather resistance, wrap gray tape on the outside. However that just adds more material and cost.
It's my understanding that Yuengling lager cartons make the best sleeves. Keep a supply and share with friends. Yes, the beverage and the sleeve!