  A closed-form formula is derived for the number of occurrences of matches of
a multiset of patterns among all ordered (plane-planted) trees with a given
number of edges. A pattern looks like a tree, with internal nodes and leaves,
but also contain components that match subtrees or sequences of subtrees. This
result extends previous versatile tree-pattern enumeration formulae to
incorporate components that are only allowed to match nonleaf subtrees and
provides enumerations of trees by the number of protected (shortest outgoing
path has two or more edges) or unprotected nodes.
