We described the probability distribution of the PWIT relative to its root. The PWIT has a `stationarity" property, saying that if you re-root at a vertex v about which you know nothing then the tree, relative to this new root, has the same distribution as the PWIT. Unfortunately, when you actually do a sequence of clicks to re-root in our simulation, you are inevitably using some information about the tree to decide where to click, and this leads to some bias which is hard to quantify, but where conceptually the tree rooted at a vertex v is distributed as the PWIT, conditional on the information used to choose v.
Here is the simplest illustration.
The shortest edge at the root of the PWIT has Exponential (mean 1)
Suppose we re-root at the other end of that edge.
In the re-rooted tree, what is the distribution
of the shortest edge at the root?
Well, the shortest edge is the shorter of
(a) the edge from the original root
(b) the shortest edge to children
and these are independent random variables each with Exponential (mean 1) distribution, so the length of the shorter edge has Exponential (mean 1/2) distribution.
Even if you click on a random vertex within the opening window,
there is a bias; the re-rooted tree will tend to have more
vertices within its window than does the true PWIT.
You can try the following experiment.
(a) Keep clicking get another realization and count the number of vertices within the window each time.
(b) Fix one realization and click around the tree, counting the number of vertices within the window each time.
In the first case, the average number will be around the theoretical value 20 mentioned on the last page. In the second case the mean should be rather larger.