Should any special effort be made to target these specifically as a group? Or should you just go about everything in order without worrying about it?

Going for them as a group, I guess, would make them even harder (since you'd be learning similar words one after another). Personally, I still mistake about half of them, but considering that in my native language, I can't recognize most trees in the wild even if I tried, I don't think it's much of a miss to just remember they're trees, paying little attention to particulars unless you are familiar with that specific tree sort (and if you are, stories should come that much easier).

I looked up all the trees that didn't sound familiar as I encountered them... once I saw the lists of common names on the wikipedia pages, there weren't any that I was totally unfamiliar with. Plus there's all that information about the tree if it's not very familiar once you see the other names the tree goes by.

The tree kanji are mostly used for people's names.

So, in practice, you're more likely to remember their
pronunciation than their actual meaning.

In all honesty, to most people "a tree is a tree".
Even if I knew the specific name of a tree, it probably
wouldn't make a difference to most people.

It's fairly useless to memorize a lot of the tree kanji. Tree names are often written in katakana anyway, and even when they appear in kanji they get furigana fairly often. As chamcham said, a lot of them are more important for their appearance in names than for their actual tree meaning.

Fadeway Wrote:Going for them as a group, I guess, would make them even harder (since you'd be learning similar words one after another).

Well, I didn't mean learning them out of sequence, just making an extra effort on top of the normal process to go over the ones you've learned.

But the other points made are well taken.

I know a lot of tree names in Hebrew (my native language) that I have no idea how they look like. I also know a lot of tree names in English that I have no idea about their exact meaning or how they look like. I'd be surprised if it actually matters at all.

I skipped tree names in English and I also skipped the RTK 3 trees. I don't see the point in learning them.