The fact that acyclicity corresponds to having unique paths powers a lot of useful arguments in various areas of mathematics. What is the most fundamental reason you can come up with to explain the correspondence?

Also, what are more sophisticated generalizations of this correspondence? By this, I mean connections between negative and positive structural properties of an object, which in some way, perhaps only informally, generalize the fact about acyclicity. I'm looking to collect examples.

Please feel free to close this question if it's deemed too vague or philosophical for MathOverflow.

undirectedgraphs, right? – François G. Dorais♦ May 7 '10 at 3:47