These are two part terms - where the endings are usually left off:
make like a banana
make like a tree
The endings though are where the point lies:
(make like a banana) and split
(make like a tree) and leave
Wikitonary just tags these as idiomatic.
Is there a technical term for these phrases?
Edit: does it count as an aposiopesis?
This is an example of ellipsis:
Ellipsis: A situation in which words are left out of a sentence but the sentence can still be understood (Cambridge Dictionary)
When written down, uncommon phrases would be appended by '...', for example: 'he pulled the trigger...'. But since the expressions you are using are idiomatic, this is not necessary, because they have formed a distinct phrase in language. Examples of this are cockney rhyming slang, where superfluous words are left off when they become phrases, eg:
Loaf of Bread: Head
Think about it; use your loaf. (fun-with-words)
In the above case, the 'of bread' from the original phrase is ellipted, because it is so common. Interestingly, 'loaf' doesn't rhyme with head, as the original phrase 'loaf of bread' did.
These are jokes--plays on words. "Make like a bee and buzz off." "Make like an egg and beat it." There's a whole list, funny, not so funny, in poor taste, and so forth at http://www.netfunny.com/rhf/jokes/91q4/makelike.html. The second part of the joke is something the smart kids can figure out--e.g., "Make like a prom dress and ..... "take off", of course.
When someone omits something but you derive the meaning from context, that is just an implied meaning. The technical term that you imagine either does not exist or I do not know it. Moreover this is just an example of an idiom with part of it left unsaid; an implication.
For example When in Rome is a popular saying. Some people may have forgotten that it used to be commonly punctuated with do as the Romans do. The verb phrase is derived by the user from context.
While the answer by @marcellothearcane is correct in that the leaving off a portion of the quote would be termed ellipsis, there's another portion of the construction that the two share (which rhyming slang doesn't).
I'm specifically referring to the wordplay part, e.g. where "to leave" as in "to depart" is put in the same context as "to produce new plant leaves" and "to split" as in "to depart" is put in the same context as "the dessert banana split".
This would be an example of zeugma:
zeugma (n): A figure of speech in which a word applies to two others in different senses (e.g. John and his driving licence expired last week). (Oxford Dictionaries)
This is also sometimes called syllepsis
syllepsis (n) A figure of speech in which a word is applied to two others of which it grammatically suits only one (e.g. neither they nor it is working). (Oxford Dictionaries)
The Wikipedia page on zeugma and syllepsis attempts to classify and clarify the different types of zeugma and syllepsis, as different sources draw slightly different distinctions between the two. They all share the characteristic, though, of a parallel construction where the two parts don't quite match up.
(I am unaware of a specific term that combines both the concept of ellipsis and zeugma in this way.)
