Originally posted by McCloud:

Birch seeds are not a thing.

Yeah, this. No tree that yields fine wood drops seeds. Fine wood is non-renewable. Other posters may have been getting birch confused with beech, which drops plenty of seeds.

Originally posted by Marimo7:

When it comes to trees that you grow yourself, the stage of growth is relevant. Even if the tree appears to be fully grown, there is another stage, and when the tree is cut down in the previous stage, the seeds will not fall out.

I've been farming trees for a while now and I cannot confirm this. Trees only seem to have two states, the tiny saplings that you initially plant and the giant full-grown trees that they explode into once enough time has passed. The latter state appears to drop seeds normally (which is to say, not all the time, but for beech trees often enough to yield a surplus on average).