Do trees have no set lifespan and
"quite"immortal?
There are certain annual(eg.peas) and biennial plants(eg.carrot) also which die at fixed times.

But for perennial plants :

From this


  Trees endure as long as they do basically because they're
  nonhierarchical organisms. In animals, all vital functions are
  controlled by the central nervous system, the guiding element of which
  is the brain. When the brain dies, so does the animal. By contrast,
  vital functions in trees are decentralized. A large part of the tree
  can die, and indeed routinely does die, without killing off the tree
  as a whole. Most of a mature tree is dead except for a few layers
  under the bark.....
  
  All trees die eventually, of course. Four thousand years is old
  compared to the life spans of gossamer creatures like ourselves, but
  in the context of geologic time it's the blink of an eye. As they get
  older trees become more susceptible to disease, pests, and other
  perils, and inevitably these take their toll.