Taxonomy term view with number of items for parent and child terms
I now have worked out a solution. You need to implement the taxonomy edge module. With that you can define a new relationship: Taxonomy term: child term. After that it's possible not only to count the number of nodes tagged with one term, but also the number of nodes tagged with the child terms of this term.

My nodes are always tagged with terms from the deepest level. So I have to consider the number of nodes tagged with the considered term OR the number of nodes tagged with the childs terms of the considered term (if available).

Completing my initial post, I add two relationships:

Taxonomy term: Child terms (Identifier = Childs)
(Childs)Taxonomy Term: Content with term (Identifier = node_childs)


I further add one field:

(node_childs) COUNT (Content: Nid)


In the field settings, i set these fields inline:


Taxonomy Term: Name 
(node) COUNT (Content: Nid)
(node_childs) COUNT (Content: Nid)


The field (node) COUNT (Content: Nid) is excluded from display.
In the field (node_childs) COUNT (Content: Nid) I set the No-results-behavior to the token [nid]. 

The effect is that either the considered term has child-terms, then the number of nodes tagged with the child terms is counted. Or the considered term has no childs, than the number of nodes tagged with the considered term is counted.