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Bananas are one of two different fruit that grown on Palm Trees. They do not occur natively in towns. These tropical fruit sell for 250 Bells at Re-Tail and 200 Bells at Nookling Store.
HOW TO GET Bananas IN ANIMAL CROSSING: NEW LEAF
You can get bananas by visiting the beach in your friends' villages, receiving them as gifts from other villagers, or traveling to the Tropical Island and harvesting them yourself. You can plant bananas only on the sandy beaches of your town -- they will not grow in the town itself. Unlike Fruit Trees, banana trees only carry two fruit at once.
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Perfect Bananas
You cannot grow perfect bananas in Animal Crossing: New Leaf. See: Perfect Fruit.