What are applications of alphabetic trees?
You use alphabetic trees when you need nodes to keep their input order in the resulting code. Huffman trees are used for coding purposes, and alphabetic trees for storing data and for allowing quick search and edit.

One example application is storing textual data in alphabetic order. The complete works of Shakespeare were stored in an OAT (optimal alphabetic tree), which was constructed using the Hu-Tucker algorithm. See here for another $O(n \log n)$ algorithm for constructing OATs.