@Rewton: Herman raises a good point (about one side dead, one side alive). It's very good advice (and always a good idea to listen to Herman, with all that he knows. :-). The only exception to that that I have practiced, is when there's a tree where the only trunk (or all of the trunks, as the case may be) is half dead and half alive. In that case, if I've got a trunk that is half dead and half alive, I don't cut it back to the ground and wait for a new shoot. Instead if I'm getting any green activity, I let it grow. And then later in the year, I either air-layer it or take a cutting. The mother tree will never be really healthy, but it provides a way to keep that variety around long enough to get a new tree. With the trunk of the mother tree (the trunk that's half dead and half alive), then after I've taken whatever cuttings I'm going to take, if it has a growing tip down low enough, I cut it off above that low green shoot (that's coming out of the half-live side), and I repot the mother tree deeper, burying all of the half-dead trunk (plus a little of the new shoot).
I had this happen with a Triano Calabrese last year. It was a pretty small tree to start with, and the whole tree got that half dead half live state. I ended up with two "new" small trees... one was a green cutting that I rooted (which had grown out of the live half of the trunk, until I cut it off and rooted it), and the other was the stub of the mother tree with one new green shoot on a half-live trunk (with the entire half dead trunk buried). I'm not sure how much the old roots stuck around after the repotting, but they at least gave it a start until new roots formed. Both trees took off.
Mike central NY state, zone 5a