What is the best way to remove one of the baby's? My Mom really wants one.
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Water the tree well. Then just take a sharp spade and slice vertically down into the rootball. You should try and get some roots with the sucker. You can trim up the roots if you need to. Transplant immediately, give a drink with seaweed or kelp extract and you're done. It should be no problem. Bananas are pretty bulletproof. If you want fruit from your plants, you're best off having only one sucker allowed to grow to maturity. So remove all but one. The ones with the pointed leaf are fruit bearers. The rounded leaved ones don't do so good.

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The secret is the soil.

At what temperature do we need to pull it out of the ground?
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They don't like freezing temps, so if you are talking about storing the stool until spring, do it before the first sub zero temps or frosts. Then cut off the sucker in pre-emergent spring when the temps get real again. I'm not in a cold area, so not sure about the exact timing. Bananas are tropical plants.

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The secret is the soil.