open_reference and de_novo picking are a little different - you are trying to estimate OTUs on the fly. In those scenarios you would use the tree generated by those methods.
With closed_reference, you are picking OTUs against a reference database. Since that reference database already has a high quality tree bundled with it, you don't need to rebuild the tree.
So yeah, when you are doing closed ref picking, you can just download that tree, and input that tree for your alpha and beta diversity analyses.
Best,
Jamie