Here is the simplest illustration.
The shortest edge at the root of the PWIT has Exponential (mean 1)
distribution.
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.