In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For
that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent
the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the
expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is
asymptotically given by 8πα3/2log(2)(β−1)β(β2β)242β−αlog(α)(2β−1)(α+1)(α+2)(α−12α+1) provided that α and β
grow in some fixed proportion ρ when α → ∞ . A similar result is shown for trees with α
internal nodes but with an arbitrary number of keys