When a cat is "stropping", like this fine fellow here demonstrates, are they really sharpening their claws? Well, sort of. Firstly, they strop to mark their territory and advertise to potential mates, by depositing pheromones from their interdigital glands, as well as visually marking the tree.
But what about claw sharpening? Scratching your claws against a tree (or expensive couch) will as likely dull your claws as sharpen them. But the way a cat's claws grow, as most cat owners probably guess from finding the sheds,
is like a snake's skin. A new claw grows inside the old claw, and when ready, the dull claw sheds off, leaving the sharp new one ready for action. Stropping will often help to shed these old claws, thus "sharpening" the claw, albeit in a round-about fashion.
