  Rooted trees with probabilities are convenient to represent a class of random
processes with memory. They allow to describe and analyze variable length codes
for data compression and distribution matching. In this work, the Leaf-Average
Node-Sum Interchange Theorem (LANSIT) and the well-known applications to path
length and leaf entropy are re-stated. The LANSIT is then applied to
informational divergence. Next, the differential LANSIT is derived, which
allows to write normalized functionals of leaf distributions as an average of
functionals of branching distributions. Joint distributions of random variables
and the corresponding conditional distributions are special cases of leaf
distributions and branching distributions. Using the differential LANSIT,
Pinsker's inequality is formulated for rooted trees with probabilities, with an
application to the approximation of product distributions. In particular, it is
shown that if the normalized informational divergence of a distribution and a
product distribution approaches zero, then the entropy rate approaches the
entropy rate of the product distribution.
