we all know that we age because at the end of each cell division, telomere in the chromosome shortens and we die eventually. so do we have telomere shortening in plants?
It is wild speculation that we humans die because our telomeres become short. But to answer your question yes in plants telomeres also shorten although I have not given this a thorough search.. This is a first paper to start with: http://m.pnas.org/content/92/21/9555.short
there is also a telomerase knockout paper in arobidopsis