  Probabilistic context-free grammars, commonly used to randomly generate
trees, have been well analyzed theoretically and have found applications in
various domains. Despite their utility, the distributions that the grammar can
express are limited to those in which the distribution of a subtree depends
only on its root and not on its context. This limitation becomes a challenge
for modeling various real-world phenomena such as natural language. To overcome
this limitation, a probabilistic context-sensitive grammar is introduced, where
a subtree's distribution depends on its context, and its statistical properties
are explored. Numerical analysis reveals that the distribution of a symbol does
not exhibit a qualitative difference from that in the context-free case, but
mutual information does. Furthermore, a metric is introduced to directly
quantify the breaking of this limitation. This metric is zero in the
context-free case, and is applicable to an arbitrary distribution of a tree.
Measuring this metric enables the detection of a distinct difference between
probabilistic context-free and context-sensitive grammars.
