Trees can adopt a range of strategies not available to animals in order to live for very long periods of time, but they are not immune to mechanisms of aging. That said, those mechanisms are only broadly similar to the biochemistry of aging in animals. It isn't clear that there is anything useful to learn from long-lived plants insofar as human medicine is concerned. Nonetheless, it is an interesting area of study.
The oldest trees on Earth have stood for nearly five millennia, and researchers have long wondered to what extent these ancient organisms undergo senescence, physically deteriorating as they age. A recent paper studying ginkgoes, one of the world's longest-lived trees, even found that they may be able to "escape senescence at the whole-plant level," raising questions about the apparent lack of aging in centuries-old trees. However, researchers argues that although signs of senescence in long-lived trees may be almost imperceptible to people, this does not mean that they're immortal.
"When we try to study these organisms, we're really astonished that they live so long. But this doesn't mean that they're immortal. They live so long because they have many mechanisms to reduce a lot of the wear and tear of aging. They have limits. There are physical and mechanical constraints that limit their ability to live indefinitely." However, due to the difficulty of conducting research on trees with such long lifespans, little is known about what the process of senescence looks like. Simply finding enough millennial trees to study can be challenging. "When a species of tree can live for five millennia, it's very difficult to find even two trees that are between two and five millennia." For these long-lived trees, dying of senescence is a possibility, but the probability of dying from other causes is significantly higher.
Trees have a variety of ways to reduce their chances of death from aging alone, from compartmentalizing risk in complex branch structures to "building life on death" by growing new shoots from trunks composed of 90% nonliving biomass. But researchers maintain that even though long-lived trees can survive for millennia through these methods, the stress associated with aging, although little, will ultimately prevent immortality.