Another term that is not intuitive is "swamped barrel". A swamped barrel is full diameter at the breech, tapers down to a narrower diameter half way along the barrel, and then flares back out to larger diameter at the muzzle.
The reason I have been told for swamped barrels being made is balance. They do indeed balance better offhand than a "straight" barrel with no taper. The breech is full to take the pressure. The swamp is thinner to relieve weight (especially out front) overall. The flare adds a bit of weight to the muzzle for stability offhand.
Swamped barrels I tend to think of in full-stock guns, and most half-stock guns tended to have straight or simply tapered barrels.
I hope this helps. God Bless, Marc