Lets assume you are a geologist 100000 years from now. When you look at a drilling core you most certainly will be able to identify a strata, that is enriched with radionuclides (or fission products of radionuclides), that cannot be found deeper (earlier) in the core.
>> Wouldn't the tree eventually die and have its remains scattered around the globe?
Yes, thats exactly the thing, a lot of trees (and other biomatter) will die, and all of them will scatter their slightly more radioactive remains around the globe to create a layer as an imprint of the history of our planet.