Description:  Ever needed (or wanted) to generate a tree (in this case, a graph on n vertices with n1 edges) at random? Dismayed that the obvious method (adding a 12 edge, then randomly picking a vertex to connect 3 to, then 4) doesn't produce all possible trees (132 simply can't be done)? Your prayers are answered! This code uses Prüfer sequences (for which Google generates no useful introductory hits) to describe trees... turns out you can rank the n^{n2} labelled trees, generate the Prüfer sequence corresponding to that rank, then reconstruct the tree from the sequence. This code is pretty much a straight implementation of the pseudocode in Combinatorial Algorithms: Generation, Enumeration, and Search. 
use strict; use POSIX qw(floor); use Data::Dumper; # rank_to_prufer(r, n) # Generates the Prufer sequence for the nvertex tree of rank # r. sub rank_to_prufer { my ($rank, $n) = @_; my @prufer = (); for(my $i = $n  2; $i > 0; $i) { my $new = $rank % $n + 1; unshift @prufer, $new; $rank = POSIX::floor(($rank  $new + 1) / $n); } return @prufer; } # prufer_to_tree(seq) # Generates the tree corresponding to the Prufer sequence seq sub prufer_to_tree { my @prufer = @_; my $n = scalar @prufer + 2; my @edges = (); my @degrees = (1) x $n; push @prufer, 1; for (0 .. $n  3) { my $i = $prufer[$_]  1; $degrees[$i]++; } for (0 .. $n  2) { my $x = $n1; while($degrees[$x] != 1) {$x;} my $y = $prufer[$_]  1; $degrees[$x]; $degrees[$y]; my @edge = ($x+1, $y+1); push @edges, \@edge; } return @edges; } # example call  generate a random 6vertex tree my @tree = &prufer_to_tree(&rank_to_prufer(rand(1296), 6));


Replies are listed 'Best First'.  

Re: Random Trees
by dmitri (Priest) on Feb 11, 2003 at 20:16 UTC  
by FoxtrotUniform (Prior) on Feb 11, 2003 at 21:10 UTC  
Re: Random Trees
by prashantpokhriyal (Novice) on Sep 30, 2017 at 15:29 UTC  
by Anonymous Monk on Sep 30, 2017 at 18:20 UTC 