If there wouldn't be other constraints, I believe I'd go rather another way, towards something like

use strict;
use warnings;

my $str = 'aabcdabcabcecdecd';
my $min_length = 1;
my $min_count  = 2;

my %count;

count( $str );

while (my ($string, $counts) = each %count) {
    if ($counts >= $min_count) {
        print "$string => $counts\n"
            if length($string) >= $min_length;
    }
}

exit;

sub count {
    my( $string) = @_;

    my $length = length( $string );
    return if $length < $min_length;

    for my $l ($min_length..$length) {
        my $substring = substr( $string, 0, $l );
        $count{$substring} += 1;
    }
    count( substr( $string, 1 ) );
}
[download]

Which gives a result that slightly differs of your expected values, but I believe is a accurate:

ec => 2
ecd => 2
ab => 3
abc => 3
bc => 3
cd => 3
[download]

and respectively

e => 2
a => 4
ecd => 2
ab => 3
c => 5
abc => 3
bc => 3
cd => 3
d => 3
ec => 2
b => 3
[download]