Syntactic Confectionery Delight  
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Re(2): Random Treesby FoxtrotUniform (Prior) 
on Feb 11, 2003 at 21:10 UTC ( #234531=note: print w/ replies, xml )  Need Help?? 
That was actually the first thing I googled for. Apparently, though, "Prufer" is the more common Anglicism, and neither produced any useful introductory documents. So what the hell, I'll give it a shot. The idea is to generate a unique sequence for each labelled tree on N vertices. What you end up doing is removing vertices one at a time, starting with the highest label leaf vertex (you can start with the lowest, it doesn't matter, as long as you're consistent). When you remove a vertex, you add the label of the vertex it was adjacent to to the sequence. Keep going until you only have two vertices left. You've got a Prüfer sequence! So, for example, if you have the tree:
You'd start by removing vertex 6 (highest label leaf vertex). Add 3 to the Prüfer sequence (which is now 3), and you have the tree:
Now remove 5 and add 3, you have (3, 3) and the tree:
Remove 4, add 2, you get (3, 3, 2)  eventually, you end up with (3, 3, 2, 3) and the tree 13. It turns out that there's a onetoone correspondence between labelled trees and Prüfer sequences, so each Prüfer sequence uniquely determines a tree. (Which is why the code I posted earlier works.) Going from a Prüfer sequence to a tree is left as an exercise for the reader. :) 
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