What type of wood is the rarest?

As SF commented, all trees can be farmed, so once you have found at least one of each, it shouldn't matter it's rarity, but rather, how easily you can replant them. I didn't find anything about their rarity in the wiki, anyway. It might be possible to check the list of every biome in which they can generate, and check the rarity of those biomes instead. But it won't matter anyway once you have a few saplings of each type.

Regarding the difficulty to farm them, I'd argue that the hardest one is, by far, the dark oak.

Oak, spruce, birch, jungle and acacia can be grown with just one sapling. And unless you decide to collect all the leaves with silk touch, you're almost guaranteed to get several saplings back from each chopped tree. So, they're self-sustained farms.

Dark oak, on the other hand, is the only tree that can only be grown with 2x2 saplings. Spruce and jungle also have that option, but it's a choice: 1 sapling for a regular tree, or 2x2 for a humongous, giant one. With dark oak, it's either 2x2 or nothing.

Also, from personal experience (in Bedrock edition), you don't get as many saplings back. If you break by hand every single leaf block, you get between 3 and 5 saplings back. Maybe 6 if you're lucky. But you invested 4 to grow the tree, so... it can still be self-sustained, but you have to be careful not to let any saplings despawn. Which could easily happen if you're cutting down a lot of trees at a time.

As SF pointed out, breaking the leaves with a Fortune III hoe will increase the amount of saplings you get back (you get exactly twice as many), making it easier to keep growing trees. But it's still extra work that you don't really need to do with any other type of tree.