Supposing an optimal nutrient rich environment and no parasites/diseases - can a tree (whatever the species) live indefinitely? Can it die of old age?
We would need a botanical cytologist to resolve this question, but here’s a quick-and-dirty answer.
First off, a “tree” is any plant that has grown a single (or, rarely, a few) vertical trunk(s) of wood, as opposed to a bush which is woody but “low-slung” or a shrub or herb which is not (or only slightly) woody. At least three different classes of plant have species which grow into trees, so the answer is going to vary from plant to plant across a broad range of flora.
That said, a good generalization is that plants do not “age” as higher animals do, but rather continue growing and flourishing while nutrients are available to support them, but begin to age and die when toxins or lack of nutrients impede that growth.
A few plants will live for a specific period of time and then die – annuals in horticulture are obvious examples, and bamboo has a specific life span. Whether any true trees fall into this group, I don’t know.
I’ve seen stuff that suggests that the “normal life span” for elms and for oaks is around 300 years, though whether this pertains to a telomere-based aging process or simply to how long an elm or oak can continue to grow healthily in one place, I don’t know.
The conifers hold the records for age, and appear to have life spans indefinite for all practical purposes. Everyone knows, I think, about the ancient bristlecone pines in that one site in California or Nevada that, while badly abused by a particularly stressful environment, have survived since around 2000 BC. But more importantly, the Sequoiaceae – the redwoods and Big Trees – regularly flourish for hundreds of years, with a few of the oldest Big Trees apparently in healthy maturity and only a few decades younger than the bristlecone pines. And you will find that references to the oldest trees in individual states and nations are very often to some conifer.
This question was apparently so important that Cecil answered it twice. Please note that the version with italics is also printed in Triumph of the Straight Dope as question four in the chapter “Natures’ Way”.