Well, neither really. Like Truffles, mushrooms fall under the category of fungi. This is because truffles are subterranean mushrooms (Fungus Tuber Ascomycetes Mycota - or sac mushroom to you and me).
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Contrary to stubborn legends, truffles no longer elude techniques of domestication. It was true historically, However as early 1808, there were successful attempts to cultivate truffles, known in French as trufficulture.
Men had long observed that truffles were growing among the roots of certain trees, such as oak trees in particular, and indeed scientific research has proven that the truffles live in symbiosis with the host tree. In 1808, Joseph Talon, from Apt (in southern France) had the idea to sow some acorns collected at the foot of oak trees known to host truffles in their root system. The experience was successful: years later, truffles were found in the soil around the newly grown oak trees.
In 1847, Auguste Rousseau of Carpentras (in Vaucluse) planted 7 hectares (17 acres) of oak trees (again from acorns found on the soil around truffle-producing oak trees), and he subsequently obtained large harvests of truffles. He received a prize at the 1855 World's Fair in Paris.