http://qs1969.pair.com?node_id=234289
Description:

Ever needed (or wanted) to generate a tree (in this case, a graph on n vertices with n-1 edges) at random? Dismayed that the obvious method (adding a 1-2 edge, then randomly picking a vertex to connect 3 to, then 4) doesn't produce all possible trees (1-3-2 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 nn-2 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 n-vertex 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 = $n-1;
    while($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 6-vertex tree
my @tree = &prufer_to_tree(&rank_to_prufer(rand(1296), 6));