How to calculate total children count for all parents from a Oracle tree hierarchy?

I have an Oracle Tree hierarchy structure that is similar to the following table called Hierarchy_Tree

(PARENT,  CHILD)

with the following sample data within Hierarchy_Tree:

CHILD   PARENT  
-------------------------
AA          A                   
AB          A                   
AAA         AA                  
BB          B                   
BBB         BB                  
BBBA        BBB                 
BBBB        BBB  
C1           C
C2           C
C3           C
C4           C3
C5           C3
C6           C3
C7           C6
C8           C6

Based on the above sample all parent (top level only required), that is, I need to produce a SQL query that returns the following information, i.e.:

PARENT   COUNT
----------------------------
A       3
B       4  
C       8

So I need to rollup total children that belong to all PARENT

I tried something liked this, but it didn't work.

select child1, count(*)-1 as "RESULT COUNT"
  from (
    select connect_by_root(child) child1
    from Hierarchy_Tree
    connect by nocycle parent = prior child
    )
group by child1
order by 1 asc

select rootp as parent, count(*) as count from (
	select child, parent, connect_by_root parent as rootp from Hierarchy_Tree
	start with parent in (select parent from Hierarchy_Tree d1
	where not exists(select 1 from Hierarchy_Tree d2 where d2.child = d1.parent)
	)
	connect by parent = prior child
)
group by rootp
;


A	3
B	4
C	8