Well, for any given target node you build a string of the correct number of left and right moves (or zeroes and ones) and then any permutation is one of the admissible solutions (and all of them).

UPDATE: Just as illustration, using a module, and having to filter out duplicates in the permutations, it could work like this:

use strict;
use warnings;
use Algorithm::Permute;

my $node = '5-2';
my ($all, $right) = split /-/, $node;
my @path = ((0) x ($all-$right),(1) x $right);
my %pathes;
Algorithm::Permute::permute { 
    my $key = join '', @path;
    if( not exists $pathes{$key} ) {
        $pathes{$key} = 1;
        my ($l, $r) = (0,0);
        my $path = "($l-$r) ".join( " ", map{ "(".(++$l)."-".($r+=$_).
+")" } @path);
        print "$path\n";
    }
} @path;
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