SALEM, Mo. — Standing on a wooded hillside in the Ozarks, about 100 miles southwest of St. Louis, Brad Boswell watches a chain-saw-wielding logger make several deft cuts at the base of a 100-foot white oak. The logger points to a clearing down the slope and, with one final, quick slash, sends the tree falling, exactly where he pointed.

Mr. Boswell scrambles over to look at the swirls and loops that make up the tree’s cross section. If they’re consistent, and the wood doesn’t show scars from fire damage or disease, it will most likely end up in some of the hundreds of thousands of barrels that his 1,500-person company, Independent Stave, turns out every year.

His great-grandfather, T. W. Boswell, founded Independent Stave in 1912, and Brad Boswell now runs it with his brother and sister. Based in Lebanon, Mo., it is the world’s largest barrel manufacturer, at a time when demand for wine, whiskey and beer — all of which rely on barrels for aging — is skyrocketing.

The United States is now the largest market for wine barrels. Domestic whiskey production is up 41 percent in the last decade — and, thanks to a quirk in federal law, almost every drop has to be aged in a new oak barrel. The demand has come on so suddenly and vertiginously that barrel prices are up 70 percent since 2012, and some cooperages have 12-month waiting lists.