On the one hand, Archaeopteryx could probably blend in and just look like a bird that they put together a little wrong until you got up close and realized its actually a freaking dinosaur. I bet they had a mean bite, but alas, they were neigh but a kilogram (2.2lbs) and just a wee tad over a foot (30cm) if you stretched them out long enough.
On the other hand, Archaeopteris is a tree, and not just any tree, but one of the FIRST trees. Early Trees are the only beings besides Humans that are personally responsible for a Mass Extinction, so that could be kinda badass depending on how you look at it or just depressing. They reproduced through spores, which is really cool. Especially since their closest living relative, Clubmosses, can have their spores used as flash powder to explode stuff and magic tricks.
Its a tough call. Realistically, I could see the Archaeopteris falling and crushing the Archaeopteryx, but then they'd be lose. I dunno what kind of a diet Archaeopteryx prefers, but if they eat plants, I guess technically they could eat an entire Archaeopteris, but it'd take a shitload of them seeing as they only weigh a kilo.
Archeopteryx (Is it just me, or does its face kinda look like the sicko guy)
Archaeopteris (Its tougher to find a photograph of these guys that makes them look adequately cool, but trust me this one doesn't do them justice even though its the best i could find)
this is unnecessary infighting, OP is a plant
edit: possibly even an Archaeopteris
Based on the skull/teeth, archaeopteryx was almost certainly a carnivore. If you give it the benefit of the doubt and assume it would even try to eat the tree, it still wouldn't be able to consume that much food or even pierce the proto-bark.
And I'm really going out on a limb here (HA TREE PUN), but if you've ever played with clubmosses, you'll know how just shaking them a little bit will get spores going E.V.E.R.Y.W.H.E.R.E. Now, Archeopteris is pretty distantly related to clubmosses (its just that everything else its closer to is extinct as well), and I don't know if that flash powder thing only applies to certain species of glubmoss like Creeping Cedar or if its all of them. Looking at wikipedia it actually says various other fern allies like horsetails can be used to make it too. So I'm imaging Archeopteryx propping itself down against Archeopteris to have a smoke , and roughling its branches a little bit, sending clouds of spors everywhere.. It takes out a lighter, and BOOM. I remember reading way back when that the coal forests were almost never not on fire, which is what helped to ave perfect conditions for turning to fossil fuel. I wonder if thas why
https://i.pinimg.com/originals/6a/87/1a/6a871af3a7754a3ac7f1b2c11e4054bb.jpg
I have further evidence for this.
I could see the Archaeopteris falling and crushing the Archaeopteryx
Trees have tried this at times. Here's the type specimen of Parasaurolophus walkeri as an example. See the indentation on the spine? It probably got hit by a faling tree there, like this.
But Parasaurolophus was a herbivore. Able to rear up on its hind legs to graze on tree branches. Clearly an enemy of trees. Archaeopteryx, on the other hand? May have been arboreal. May have lived in the treetops getting all comfy being perched up there. Clearly carnivorous, too, had no business harming trees.
Archaeopteryx could win if you gave it an axe that could fit in its mouth and it had enough time and determination.