Renaissance is links.
Links courses can have trees. In fact, links land over years and years will actually turn in to woodland, but only as the shore accretes outside of it. Look up seral succession of sand dunes if you want more info.
That said, the inland parts of Renaissance were full of planted trees. That is a little different.
Heathland always has nutrient poor soil. Heathlands (as we think of them for golf) are almost always sand; but wet heathlands, upper heath (i.e. moorland) will not be sandy sub-soil.