The process that would lead to offspring with the exact same genetic information as the parent is a bacterium splitting in two. This process is known as binary fission, which is a form of asexual reproduction that results in two identical daughter cells, each with the same genetic material as the parent bacterium.
While a tree growing from the stump of another tree (a form of vegetative reproduction) can also produce genetically identical offspring, it is not as straightforward as binary fission in terms of genetic replication.