Botanically speaking, the tomato you eat is a fruit. So is a watermelon, green pepper, eggplant, cucumber, and squash. A "fruit" is any fleshy material covering a seed or seeds.
Horticulturally speaking, the tomato is a vegetable plant. The plant is an annual and nonwoody. Most fruits, from a horticulture perspective, are grown on a woody plant (apples, cherries, raspberries, oranges) with the exception of strawberries.
In 1893, the United States Supreme Court ruled the tomato was a "vegetable" and therefore subject to import taxes. The suit was brought by a consortium of growers who wanted it declared a vegetable to protect U.S. crop development and prices. Fruits, at that time, were not subjected to import taxes and foreign countries could flood the market with lower priced produce. (A hundred years really hasn't changed anything.)
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