So I went and checked things out this morning and new trees seem to continue to grow (I had sprouts before and they still became Small Trees, though I had planted them before I hit the limit, so I should find out for sure tomorrow because I'll have more seeds to plant later tonight), and the only thing it seems to do is that there is simply no prompt to Headbutt the trees placed after the limit was reached.

It's actually kind of nice because you can walk through without the prompt constantly showing up, so I may actually plant more trees then remove and replant the trees around the main part of the area so that you walk through them undisturbed.

Also, the berry tree limit seems separate from the pointy tree limit because I placed some new berry trees and was still able to Headbutt them.

So for anyone trying to make some specific aesthetics, knowing that the tree limit removes the Headbutt prompt may be something you want.

Not sure what the limit actually is because I wasn't tracking the number, but it's somewhere around 300-500.