Abstract
In this paper, we examine finite unlabeled rooted planted binary plane trees with no edge length. First, we provide an exact formula for the number of trees with given Horton-Strahler numbers. Then, using the notion of entropy, we examine the structural complexity of random trees with vertices. Finally, we quantify the complexity of the tree's structural properties as tree is allowed to grow in size, by evaluating the entropy rate for trees with vertices and for trees that satisfy Horton Law with Horton exponent .
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