Supporting delete of last character in a suffix tree/automaton

Is there a way to support removal of last added character from a suffix tree or automaton?

I’ve seen couple problems on codeforces and codechef (like this) in which strings in a query are formed by extending previous strings. Thus if I could decompose the strings into something like a directed tree or spaghetti stack, it should solve it easily.

It might be possible by making the tree persistent, but since the tree by itself “contains” all the information about S[0...n-2], is it possible to retreive the older tree without persistence?

I am curious about this too, but now I don’t have time to think about it. Could you @ me sunday?