  In this paper we examine planted binary plane trees. First, we provide an
exact formula for the number of planted binary trees with given Horton-Strahler
orders. Then, using the notion of entropy, we examine the structural complexity
of random planted binary trees with N vertices. Finally, we quantify the
complexity of the tree's structural properties as tree grows in size, by
evaluating the entropy rate for planted binary plane trees with N vertices and
for planted binary plane trees that satisfy Horton Law with Horton exponent R.
