Alphabetic trees are Huffman trees with the additional constraint on order, or, equivalently, search trees with the modification that all elements are stored in the leaves. Faster algorithms exist for optimal alphabetic binary trees (OABTs).
From this, I understand that an alphabetic binary tree will produce shorter paths from root to element for elements that have been inserted into the tree more times than elements that haven't, and the alphabetic property would be described the same way for a ternary tree.
I haven't been able to find a lot of discussion about applications for the alphabetic property outside of Huffman coding but it is not a requirement to implement the algorithm. Are there any applications where the alphabetic property would be a requirement, and if not, are there any benefits to guaranteeing the alphabetic property that justifies increased implementation complexity?