Note that a point is a tree, and a pair of points is a forest.  Thus with the given definition the circle $S^1$ is a two-tree. (In fact, any finite graph is a two-tree with the given definition.)  Thus all surfaces are three-trees, and more generally all $n$-manifolds are $(n+1)$-trees.  

The examples given in the original post strongly suggest that this is not the desired outcome.  Perhaps the following definition is closer to what you want. 

 1. The *zero-tree* is the one-point space. 
 2. A connected topological space $X$ is an *$(n+1)$-tree* if for any pair of distinct points $x, y \in X$ there is an $n$-tree $Y \subset X - \{x, y\}$ separating $x$ from $y$.

I think that life is nicer if we also add the assumption that $n$-trees are homeomorphic images of CW-complexes.  With this additional assumption I believe we can prove the following.

> **Lemma**: Suppose that $X$ is an $n$-tree.  Then $X$ is simply connected.

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**Edit:** As Gera points out, the annulus is an easy counterexample to my claimed “Lemma”. I now do not see how to strengthen the OP hypotheses without making them very combinatorial… If you are willing to do that, then Ian’s answer seems like a very natural direction to go in.