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.



Acyclicity equivalent to unique paths
One possible generalization comes from the graph minors project of Robertson and Seymour. In particular the notion of [tree-width](http://en.wikipedia.org/wiki/Tree_decomposition) is in some sense dual to the notion of a *bramble* (I will define this in a second). Note that a connected graph has tree-width 1 if and only if it is a tree.


Now, let $G$ be a graph. Two subsets of $V(G)$ *touch* if they have a vertex in common or $G$ contains an edge between them. A set of pairwise touching
connected vertex sets in $G$ is a *bramble*. A subset of vertices *covers* a bramble $\mathcal{B}$ if it intersects every set in $\mathcal{B}$. The least number of vertices covering $\mathcal{B}$ is the *order* of $\mathcal{B}$.


Here is the duality relation that I alluded to earlier. 


**Theorem.** A graph has tree-width $< k$ if and only if it does not contain a bramble of order $>k$. 


So, loosely think of tree-width as a generalized unique paths property and brambles as generalized cycles. 


