I might be missing something obvious, but don't shotguns fire shot-filled cartridges? Shotgun cartridges leave a metal base and a plug of some sort of wadding (looks a bit like cardboard) behind when the gun is fired, along with a bit of the plastic outer case, but those don't get fired: only the lead shot does. So you wouldn't be digging a single slug out of the tree if it was fired there by a shotgun: you'd be digging lots of little lead pellets out of the tree.
At least, that's how our shotguns worked.
That is normal for most bird hunting, OH, as quicklime mentioned. You are indeed correct, a shotshell is a an empty cartridge filled by first a powder charge, then plastic wad, filled with shot, then crimped. A primer is in the base, which the firing pin detonates. The shot, wad and burned powder fire outward, leaving the empty cartridge to be either discarded, or in my case, to be loaded once again. Simply push out the fired primer, add powder, wad and shot, crimp and you're back in business, for brevities sake.
In much of the eastern US, shotguns are the only legal weapon to use for deer and bear hunting. The area is more populated and high powered rifle cartridges will fly up to 3 miles if there is no backstop. To lengthen distances from 40-50 yards with traditional double-ought buckshot, slug technology has leapt ahead by leaps and bounds. No longer is it simply driving a pure lead slug down a smooth bore with a bead front sight and no rear. Now there are rifles slugs, rifled bores, using rifle sights. While they may sound like going the wrong way, it really isn't.
A hunter wants as clean a kill as possible. To be able to place the projectile as perfectly as possible into the kill zone, the rifled slug makes it that much more precise. Add a rifled barrel to any slug and it is yet more precise. Now add good adjustable rear sights, or a scope and you have a firearm that is still short ranged, perhaps 125-150 yards, yet is capable of fine accuracy. Incredible accuracy, actually.
In Alaska and many parts of Canada, Slugs are considered the best self-defense cartridge for dangerous game. Fisherman and hunters who are in constant close proximity to both brown and grizzly bear are recommended by Alaska F&G to use pump shotguns and slugs, specifically Brenneke slugs. They deliver over 3,000 foot pounds of energy at the muzzle.