What is the name of this part of the plant, fruit, vegetable? The thing that the plant is connected with the tree and gets nutrients with? The part we usually cut out when eat fruit.
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'Stalk' or 'pedicel' would be an appropriate term (see, for example, this paper or this one). Specifically, you could say 'terminal part of the stalk/pedicel', though I don't know if there is a word for that.
Note that the term pedicel is commonly used for the stalk of a flower; it makes sense to use it for fruits too as they are derived from flowers.