The Kaftor Va’ferah (work by Ishtori Ha’parhi, 14th century) maintained that it is forbidden to eat eggplant – a plant which grows on trees that dies within three years after it grows. He reasoned that since the Torah forbids eating Orla – the fruit of a tree produced during the first three years after the tree is planted – it is forbidden to eat the fruit of a tree that dies before the end of three years.
This position was disputed by the Radbaz (Rav David Ben Zimra, Egypt, 1479-1573), who writes in one of his responsum that when he arrived in the Land of Israel, he saw prominent Rabbis eating eggplant, proving that it is permissible for consumption. He added that the eggplant tree does not qualify as a "tree" in the Halachic sense, and so the Beracha over eggplant is "Boreh Peri Ha’adama," and not "Boreh Peri Ha’etz." The reason, as explained by the Halachot Ketanot (Rav Yisrael Yaakob Hagiz, 1620-1674) based on a passage in the Tosefta, is because a defining characteristic of a tree is a slow, gradual process of producing fruit. The eggplant tree produces the eggplants very rapidly, within a year of being planted, a quality which sets it fundamentally apart from ordinary trees. It therefore does not qualify as a Halachic "tree," and, as such, it is not subject to the prohibition of Orla, and the Beracha over its fruit is "Ha’adama," and not "Ha’etz."
This ruling is corroborated by a story told by the Hida (Rav Haim Yosef David Azulai, 1724-1806), in his work Birkeh Yosef (Yoreh De’a, 294), of the three towering sages in Safed – the Arizal, Rav Haim Vital, and Rav Moshe Alshich – who once approached Maran, Rav Yosef Karo (author of the Shulhan Aruch) and inquired about the Halachic status of the eggplant. Maran replied that it may be eaten, and the Beracha recited is "Ha’adama." This is, indeed, the Halacha.
Summary: Even though the eggplant grows on a tree, its Beracha is "Boreh Peri Ha’adama."