Information gets into tree rings and ice cores by natural means. Information also gets into DNA by natural means.
Information “Got” into tree ring cores by a natural, rhythmic, cyclical process (the change of seasons). Random events did not “create” the information.
There is a always a non-random process behind the creation of information.
You might be right, if you would say that the weather is a completely random event - but we know its not. So therefore the “information” is not created, rather the tree is just recording the environment - no more than a book “writes” or “creates” the words recorded in it’s pages.
Moreover, tree rings will “never” be decoded to provide proofs, or poems, or record any information or knowledge not related directly to the growth of the tree. Compare this to a book, that can contain any idea, thought, formula, or whatever. Clearly it was *not* just a process that produced the information content in the book.