An alien is a species that is introduced in an area where it did not occur naturally.
That’s it. A simple definition (although it is only one in a million) and easy to work with! If a plant grows somewhere where you could not find him before, you may call him ‘alien’.
Unfortunately, there are plenty of problems with this simple definition. As always, nature is too stubborn to fit in the nice scientific framework that we want to press on it.
Just one example: how do you know a plant was absent from an area before?
Many aliens were introduced together with our western way of agriculture, which is somewhere from the 16th century onwards. But data from this period are horribly scarce (going back more than 30 years is already a little disaster!). We rely on the lists of some lonely historic plant freaks, who marked down the species they saw in their gardens. But this kind of freaks where scarce, and they definitely failed in writing down everything that grows!
That is why the field of Paleobotany is so extremely important in keeping the rest of ecology smoothly on the rails. They make their way through the piles of dusty historical data to reconstruct the history of all species.
There is a wonderful website that collects global historic occurrence data. I like to choose some of my study species and watch how they slowly take over the world. (Follow here the dandelion, for example).
I especially like how these dandelions take over New Zealand, Japan and all these remote islands in the middle of the ocean. Check also the huge expansion in the Americas (where it should NOT be!). The chance that they flew on their own little wings all the way to Hawaii is very unlikely. There is a much larger probability that we, humans, are involved.
There comes a small warning with the use of this kind of data: although it beautifully shows the presence of the plants, it is extremely difficult to prove its ABSENCE. As you can see, data from before 1930 are quiet scarce, not because dandelions were a rare species, but just because nobody wrote its occurrence down! We should thus be very careful in the use of this information.