Does the galaxy look like a Web or a Tree? Suppose we have a distance cutoff $r_0$.
Let's say that any two stars in the galaxy with a relative distance below $r_0$ have a path between them.
Would the path-connected Milky Way look like a Web in which every star is potentially connected to all of its neighbors or would we form a type of branched Tree structure with junctions and bottlenecks?
I understand that for a high enough $r_0$, it will always look like a web. But my doubt is whether the standard-deviation of interstellar distances in the Milky way is such that there exists a value of $r_0$ for which the topology is unequivocally a Tree.
The process where you link random points within distance $r_0$ into a graph is known as continuum percolation. As $r_0$ increases from 0 at first there are just isolated clusters (binaries, randomly very close stars) that gradually merge. At a critical distance of a few parsecs these clusters mostly merge into one major galaxy-spanning cluster, and above that distance this cluster just gets more densely connected. The big network at the critical range has a scale-free structure of local clusters, bridges, and trees: neither a proper web nor a tree.
Some illustrations from my book manuscript:


It should be noted that these pictures assume a constant star density. In reality the stellar density varies, so for a given $r_0$ there is a part of the disk and bulge that are connected and parts of the outer galaxy and halo that are not connected.