We can put a metric on the set of isomorphism classes of connected, locally finite rooted graphs as follows:

Let $G$, $H$ be locally finite graphs, and let $u \in V(G)$ be a vertex of $G$, $v \in V(H)$ be a vertex of $H$. Define $R((G,u),(H,v))$ to be the largest number such that $(B_G(u,R),u)$ is isomorphic to $(B_H(v,R),v)$, where $B_G(u,R)$ is the subgraph of $G$ induced by the set of vertices that can be reached by starting at $u$ and traversing at most $R$ edges. We then define the distance between $(G,u),(H,v)$ to be

$$d((G,u),(H,v)) := 2^{-R((G,u),(H,v))}.$$

Let's call the topology on the set of isomorphism classes of rooted connected locally finite graphs induced by this metric the *local topology*.

Here is my question: **for which connected locally finite graphs $G$ is the set
$$ \{ (G,v) \}_{v \in V(G)} $$
compact in the local topology?**

Some observations:
It is easy to see that if a graph has unbounded degree, then the above set is not compact, so we may restrict ourselves to bounded degree graphs.
It happens that for each $D$ the set of (isomorphism classes of) connected locally finite rooted graphs of maximum degree at most $D$ is compact in the local topology. So if $G$ is any bounded degree graph, the above set is *precompact*, so it actually suffices to see whether the set is *closed* in the local topology.

If $G$ is vertex-transitive (i.e. the action of $Aut(G)$ on $V$ has a single orbit), then the above set is actually a single point, hence compact. Similarly, if $G$ is "almost-vertex-transitive", that is, the action of $Aut(G)$ on $V$ has only finitely many orbits, then the above set is finite and hence compact. (The above set being finite is actually equivalent to almost-vertex-transitivity). So the only case left is if the above set is not finite but is still compact. (Note also that since $V(G)$ is countable, the above set is at most countable; when I first thought about this I wondered if some examples could come from aperiodic tilings or something, but these examples are quickly disqualified once you realize that the set of limit points is uncountable).

So, short of a complete classification, my question is whether there are any "interesting" examples. That is:
**Is there any (infinite, bounded degree) connected graph $G$ such that the set $\{(G,v)\}_{v \in V(G)}$ is countably infinite, but is still closed (hence compact) in the local topology?**

Of course, if the answer to this question is "no" then it *is* a complete classification, and it would be quite interesting, but if the answer is "yes" I would be very interested to know any examples.