No need for sort:

use warnings;
use strict;
use 5.010;

package NotAHash;

sub new {
    my ($class) = @_;
    return bless [], $class;
}

sub add {
    my ($self, $str) = @_;
    my $parent = $self;

    $parent = $parent->[substr $str, 0, 1, ''] //= [] while length $st
+r;
    $parent->[10]++;
}


sub keys {
    my ($self) = @_;
    my @keys;
    
    for my $elementIdx (0 .. 9) {
        next if ! defined $elementIdx || ! defined $self->[$elementIdx
+];

        my $element = $self->[$elementIdx];
        my @subKeys = NotAHash::keys ($element) if @$element;

        push @keys, "$elementIdx$_" for @subKeys;
        push @keys, $elementIdx if $element->[10];
    }

    return @keys;
}


sub value {
    my ($self, $key) = @_;
    my $parent = $self;

    $parent = $parent->[substr $key, 0, 1, ''] //= [] while length $ke
+y;
    return $parent->[10];
}

package main;

my @array = qw(1 3 4 55 4 3 4 22 1 3 4 3 2 2 3 34);
my $notHash = NotAHash->new();

$notHash->add($_) for @array;

my @singles;
my @keys = $notHash->keys();

$notHash->value($_) == 1 && push @singles, $_ for $notHash->keys();
print join (', ', @singles), "  are the numbers that appear only once\
+n";
[download]

Prints:

22, 34, 55  are the numbers that appear only once
[download]