That usage of "lime" confused me for a long time. Sources I read when I was younger talked about "lime trees" being a major component in the forests of Northern Europe, and that just seemed to so very counterintuitive. (It's an odd development--lind > line > lime--linden is a recent loan from German.) Many of the lindens around here are actually "basswoods" (T. americana).

"Berry" isn't reduced for us; it has a full vowel and secondary stress. The only reduction I can think of in any of these words is /'rasp/ > /'raz/ in raspberry.