And you shall take on the first day the fruit of a beautiful tree, branches of palm trees, boughs of leafy trees, and willows of the brook, and you shall rejoice before the L-rd your G-d for seven days.
The Rebbe notes that al pi Halacha, three out of the four species that we take on Sukkos need to correspond to their description from Torah: the fruit needs to specifically esrog and the palm branches have to be from a palm (and the fruit has to be beautiful in appearance), and the 'boughs' have to be 'leafy' (they need to have at least three leaves growing from the stem).
But the branch of the willow does not have to come literally from a 'willow of a brook'. I.e., the willow tree from which we cut the branch does not have to grow by a waterway. Why is that?
It is well known that the four species represent four kinds of Jews: those that perform mitzvos, those that learn Torah, those that do both, and those that do neither; these four categories correspond to the species that have good taste, good smell, both, and neither. The willow branch belongs to the latter category.
The willow thus represents the Jew who neither learns nor performs mitzvos. And that is why the willow does not need to grow specifically by the river.
The other three species are characterized by their mitzvos and their Torah. For both mitzvos and Torah to be 'valid', they need to be connected to a tradition: we cannot perform a mitzva willy-nilly; we must have received a tradition from our ancestors as to how to perform it. The same goes for Torah learning, especially learning Halachos. While to a certain extent innovation in Torah learning is allowed and even encouraged (especially interpretation of what Chumash teaches us in avoida), even that must happen within the context of an already-existing mesoira.
That is why the species that represent the Jews whose 'claim to fame' is observance of mitzvos and learning of Torah must themselves be validated by a tradition: they must correspond to their description in Torah. (Also, for instance, our Rebbeim note that we need to have a tradition for a specific sub-species of esrog, just like we need a tradition for a species of a bird to be kosher. That is why the Lubavitchers only get Calabria esrogim from Italy or Kfar Chabad.)
But what about the fourth Jew? What's good about him? Certainly not his 'external' aspects -- he has nothing to be proud of in his service of Hashem. But we still love him, because Hashem loves him. Because he is a Jew. He has in him Essence of Hashem. And that comes to the fore when the external aspects are diminished.
That is why the willow branches do not have to be specifically from the willows that grow by the river. As long as we know that they originate from brook willows, that in their essence they belong to the species, no matter where they are right now, they can still be part of the avoida of Sukkos.