I've never attached anything, securely, to a tree without using fasteners of some kind.
But, if I needed to, I would probably use a ratcheting tie down strap for larger trunks. Or, wire wrapped around the trunk.
Maybe hose clamps for smaller limbs. (hose clamps come in some pretty large sizes and can be joined together for even longer lengths.
Assuming the intention is to not harm the tree, whatever you do might need to be repositioned every year or so.