Arman

$tree = array(
    1 => array(
        'id' => 1,
        'parent' => 0,
        'name' => 'Каталог 1',
        'count' => 0,
        'childs' => array(
            4 => array(
                'id' => 4,
                'parent' => 1,
                'name' => 'Каталог 4',
                'count' => 2,
                'childs' => array(
                    5 => array(
                        'id' => 5,
                        'parent' => 4,
                        'name' => 'Каталог 5',
                        'count' => 6
                    )
                )
            )
        )
    ),
    2 => array(
        'id' => 2,
        'parent' => 0,
        'name' => 'Каталог 2',
        'count' => 3
    ),
    3 => array(
        'id' => 3,
        'parent' => 0,
        'name' => 'Каталог 3',
        'count' => 5,
        'childs' => array(
            6 => array(
                'id' => 6,
                'parent' => 3,
                'name' => 'Каталог 6',
                'count' => 8
            ),
            7 => array(
                'id' => 7,
                'parent' => 3,
                'name' => 'Каталог 7',
                'count' => 1
            )
        )
    ),
    10 => array(
        'id' => 10,
        'parent' => 0,
        'name' => 'Каталог 10',
        'count' => 0
    )
);

$f = function ($f, &$data) {
    $count = 0;

    foreach ($data as $key => &$value) {
        if (!isset($value['count'])) {
            $value['count'] = 0;
        }

        if (isset($value['childs'])) {
            $value['count'] += $f($f, $value['childs']);
        }

        $count += $value['count'];
    }

    return $count;
};

$f($f, $tree);

header('content-type: text/html; charset=utf-8');
echo '<pre>';
@print_r($tree);
echo '</pre>';
exit(__FILE__ . ': ' . __LINE__);