  Regular tree grammars and regular path expressions constitute core constructs
widely used in programming languages and type systems. Nevertheless, there has
been little research so far on reasoning frameworks for path expressions where
node cardinality constraints occur along a path in a tree. We present a logic
capable of expressing deep counting along paths which may include arbitrary
recursive forward and backward navigation. The counting extensions can be seen
as a generalization of graded modalities that count immediate successor nodes.
While the combination of graded modalities, nominals, and inverse modalities
yields undecidable logics over graphs, we show that these features can be
combined in a tree logic decidable in exponential time.
