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, so this is a generalization of the unique paths property.  

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 $>k$.