Elaine Jackson writes
From what I understand (to save time let's agree that this qualification
applies to all of what follows), there are certain plants that have a finite
lifespan built into them: they go to seed and then they die.

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This is called monocarpic (Greek 'monos' = "single"; 'carpos' = "fruit").
***

My main question is: What about trees? I can't recall ever hearing of a
tree that died of old age.

Stewart Robert Hinsley schreef
Trees do die of old age. The life span of a tree varies, IIRC, from
about 50 years (birches, rowans) to hundreds of years (oaks) to
thousands of years (redwoods, yews, gingkos, bristlecone pines).

***
Probably there are trees with lifespans shorter than 50 years. Some tropical
trees are harvested within ten years.

Also, I recall that trees can be monocarpic also (dying after setting seed).
It is quite rare and IIRC it concerns fairly long-lived trees..
PvR