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I'm looking for any literature or results on infinite directed 1-pseudo-trees.

Brief definitions: A pseudo-tree is a tree with at most once cycle. 1-pseudo-trees have outdegree of exactly 1 everywhere. They arise naturally as components of (graph-theoretic interpretations of) arbitrary functions.

There is a few basic properties on wikipedia and tons of literature listed, but they all focus on algorithmic questions for finite trees. The same is true for anything else I found on this subject. I'm interested in learning about existing results for properties of the infinite case, especially concerning morphisms and other types of functions between such trees, existance conditions for such morphisms and similar structural results.

Lastly, for completeness' sake, this interest stems from my (far too generically phrased and thus unanswered) question on properties of composition of functions, here.

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