Summary: This post has got rather long after the discussion. The main still open Questions are 5 & 6 below. There is work in progress, and I'll post an update at some point.

A tree is a connected 1-complex $X$ every two points $x,y$ of which can be separated by a point $z\neq x,y$. (We could let $X$ be a more general topological space here.)

Going one dimension up, call a connected 2-complex (or topological space) $X$ a 2-tree, if for every $x,y\in X$, there is a homeomorph of a forest in $X - \{x,y\}$ separating $x$ from $y$. (A forest is a disjoint union of trees.)

Question 1: Has this notion been studied?

Question 2: Is it true that a 2-complex $X$ is a 2-tree iff it contains no homeomorph of a closed surface? (The forward implication is obvious.)

Some examples of 2-trees: standard trees, graphs, $\mathbb{R}^2$, and most interestingly, cartesian products $\mathbb{R} \times T$, where $T$ is any tree (to see this, embed $\mathbb{R} \times T$ in $\mathbb{R}^3$, and cut it with a plane separating $x$ from $y$).

Examples of non-2-trees: closed surfaces, $\mathbb{R}^3$, standard Cayley complex of $\mathbb{Z}^3$ (to see the latter, notice that it contains copies of $\mathbb{S}^2$).

Remark: The definition can be applied recursively, to define $n$-trees for every $n\in \mathbb{N}$.

Update 1: The answer to Question 2 is negative, as Geva Yashfe points out in the comments.

As Sam Nead points out below, the variant of the definition with “forest” replaced by “tree” seems more sensible. Now a graph is a 2-tree iff it is a tree. Defining a 0-tree to be a point, the recursive definition now starts with 1-trees being the usual trees.

The main question now is: Question 3: Is every CW-complex (or other nice space, see e.g. Ian Agol’s answer) which is an $n$-tree contractible? Simply-connected?

Update 2 The answer to Question 3 is negative: as Geva Yashfe points out, an annulus is a 2-tree.

A remaining question is:

Question 4: Is every $(n-1)$-connected CW-complex which is an $n$-tree contractible?

But here is another variant of the recurersive definition of n-tree that perhaps makes Q3 interesting again:

Definition 3:

• a point is a strong 0-tree;

• A space X is a strong n-tree, if for every two disjoint strong $(n-1)$-trees $T$, $T'$ in X, there is a strong $(n-1)$-tree $S$, disjoint from $T$, $T'$, that separates them.

Question 5: Is every strong $n$-tree contractible?

Going through the above examples and non-examples with 2-trees replaced by strong 2-trees is interesting.

My motivation goes back to a theorem of Manning, saying that a graph is quasi-isometric to a tree if (and only if) every two points $x$, $y$ are separated by a ball of uniformly bounded radius around any midpoint from $x$ to $y$. This could be reformulated as saying that $X$ is a quasi-1-tree iff every two disjoint strong quasi-0-trees in $X$ can be separated by a third strong quasi-0-tree disjoint from both. I'm looking for the right definition that will generalise this to higher dimensions. Thus I propose:

Question 6: Let $X$ be a geodesic metric space in which every two disjoint strong quasi-$(n-1)$-trees can be separated by a third strong quasi-$(n-1)$-tree disjoint from both. Then $X$ is a strong quasi-$n$-tree.

By strong quasi-$n$-tree here I mean a space quasi-isometric to a strong $n$-tree, and the constants are some function of the dimension and the constants used in dimension $n-1$.