They're supposed to be hardy to zone 4, but this was a terrible winter for some plants, in some regions. The zone 4 rating may only apply to root hardiness, too; I'm not really sure.

If you don't mind looking at the bare branches a little longer, I'd wait a while before cutting it back hard. Or, you could remove a few of the tall branches, and leave a few, to hedge your bets.

I know I've said this before, but after a particularly bad winter I once found tiny new leaves on a couple of marginally-hardy small trees in my garden on July 4, when I finally went out with my bow saw to cut them down. In my case, the plants were a crape myrtle and a vitex agnus castus, but I don't see why that couldn't happen with Cotinus.

I'm having similar questions about a lot of my hydrangeas this year; there are some very tiny leaves at the tips of the branches, and I don't know whether it's worth leaving them, or reducing the plants to the basal foliage. Hate to lose a whole season of flowers, but that might happen in any case.

Copice it. Cut it right down to just above the ground. Yes all the way down. It'll be back better than ever.

Steve

Scratch the bark. If it is a deep fleshy green below the surface, the branches are still alive. If it scratches brown, it has died back. Do this in a few spots starting from the tips down to the main trunk. That will help you determine how much of the plant has died and how much to cut back.

The other bush is showing budding a part of the way up two of the main trunks. Another trunk is completely dead. I don't think they would be budding so aggressively near the base if the top was alive. I'll do some scratching and give them a good coppicing. Thanks for the help!

Is it a purple smoke bush? IME they are a bit less hardy than the old-fashioned green type. Here at the northern end of zone 5, they often die back to the ground. You'll lose the "smoke" for this year, but the foliage will be lovely.

Both are the purple smoke bush. I've really grown to like them, as they look very different from anything else in the garden. They are both very much alive near the base of the bush as that is where the buds are starting to open up. It will be nice to see how they regenerate after I prune them. This was a tough winter here.