Yes, which doesn't make it true. The "have died eventually" applies to all participants in the battle of Trafalgar. Didn't immediately, but after some time. All dead now.

I've drunk water. I have not died.

There is no definitive proof either way. Fact is that they can live a long, long time. The oldest known trees are close to 5000 years old. There's even a clonal system estimated to be 80.000 years old. https://www.bbcearth.com/news/can-trees-really-live-forever

Fact is that a tree is much more likely to die from external factors than from old age. It cannot be proven, but it is certainly not impossible that (some) trees are immortal under the right circumstances.