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HenrikRüping
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Here is another way to generate a spanning tree in a non-uniform way. I added it as a answer, because the exercise that is non-uniform is interesting enough.

Start with the set of $n$-vertices 1,..,n. Pick a random vertex and remove it. Pick a second vertex and leave it in the set. Connect the two vertices. Then repeat until there is only one vertex left in that set.

HenrikRüping
  • 11.1k
  • 37
  • 72