使用具有指定键名的COUNT_RECURSIVE？

Question

count($arr['children'], COUNT_RECURSIVE); to get the total number of keys inside the original children key, however I'm wanting to filter out all but the keys named 'children.' I'm unsure the count_recursive works like this.

$arr =  array (
    'id' => '81',
    'parent' => NULL,
    'children' =>
        array (
          0 =>
          array (
            'id' => '173',
            'parent' => '81',
          ),
          1 =>
          array (
            'id' => '84',
            'parent' => '81',
            'children' =>
            array (
              0 =>
              array (
                'id' => '85',
                'parent' => '84',
                'children' =>
                array (
                  0 =>
                  array (
                    'id' => '131',
                    'parent' => '85',
                    'children' =>
                    array (
                      0 =>
                      array (
                        'id' => '176',
                        'parent' => '131',
                      ),
                    ),
                  ),
                ),
              ),
              1 =>
              array (
                'id' => '174',
                'parent' => '84',
              ),
              2 =>
              array (
                'id' => '175',
                'parent' => '84',
              ),
            ),
          ),
    ),
);

echo count($arr['children'], COUNT_RECURSIVE);

I'm trying to have it return 7, but because it's counting all keys, it returns 24. How could I do this?