stardew-valley – Is There Any Danger in Chopping Down Every Tree in Stardew Valley

1) As stated in a comment, there is a community centre package that requires Gold star crops. (Parsnips/Melons/Pumpkins/Corn, 5 of each)

2) In Fall, there is a festival where you will be able to show off some quality products and gain points based on their quality, so bringing higher quality items will net your more points.

3) Other than that, higher quality food will yield higher health/energy benefits.

4) I am unsure whether a higher quality items makes a "better" gift than the same item but with a lower quality, but that would be another reason to keep some good quality items.

I mostly sell the quality stuff myself and try and keep some good ones for the Fall festival.

After some testing, it does appear that non-fruit bearing trees do NOT re-grow from stumps.

This information is taken from the updated Wiki page:

Non-Fruiting trees take 10 to 12 days to grow from being planted, as long as the tiles in a two tile radius remain clear. They will grow at a reduced rate if other objects occupy part of this area and will not grow beyond a sapling if there is another tree directly adjacent. Once fully grown, the surrounding area can be paved with no ill effects on the tree, although doing so will prevent seeds from being dropped. These trees do not regrow from a stump, a new seed must always be planted.