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To my understanding, recursion is a synonym of nesting, and it's distinctive when a child clause is center-embedded in the parent clause.

I believe below is an example of recursion:
The beautiful bird, singing in the cage, is mine.

In the example above, "singing in the cage" is the child clause, and "The beautiful bird is min" is the parent clause.

What I don't know is whether a sentence like this is considered to have a recursion:
The beautiful bird is mine, and it's singing in the cage.

Like a recursive sentence, this kind of structure can be extended endlessly, but I don't think it has a parent-child relationship between the clauses.

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Yes, a simple chain of independent sentences is recursion. An interesting case of nesting recursion which seems not add to complexity is "This is the cat that chased the rat that lived in the house that Jack built."

Recursion is not really a property of syntactic constructions, but rather of grammars. The simple chaining construction you're asking about would commonly be called "recursive" because in a phrase structure grammar it would be described with a recursive rule. However, there is another device in common use, the regular expression, that can describe that pattern with an asterisk operator instead of using recursion.

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It is recursive in that the topic of the adverbial ('singing in the cage') is {technically} the verb 'be' ('is mine'), rather than the main topic (1 bird [that is beautiful]). 'in the cage' is also adverbial to 'singing', although the bird has to be at the same location. The depths would be: 0: one bird; 1: be beautiful; 1: be mine; 2: [be] singing; 3: in one cage.

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