It seems to me there is a step missing: plant seeds further north, some plants survive better than others even reproduce, take those survivors and repeat furthe

It seems to me there is a step missing: plant seeds further north, some plants survive better than others even reproduce, take those survivors and repeat further north. So not the fruit from all trees but from the ones that are better adapted. This could happen naturally but probably more slowly and also would naturally have to compete with local species.