Halin graphs contain a Hamilton cycle and have the interesting property, that, also in the case of arbitrary real edge weights, it is possible to report one of the shortest contained Hamilton cycles in $O(n)$ time ("G. Cornuejols, D. Naddef, and W.R. Pulleyblank. Halin graphs and the travelling salesman problem. Mathematical Programming, 26:287–294, 1983." or this answer ), and I would like to know, for which kind of Halin graphs the savings over enumerating all Hamilton cycles is maximal.

Question:

given the number $n$ of vertices, what is the maximal number of Hamilton cycles that Halin graphs can contain and what are examples of such extremal Halin graphs?

theshortest Hamilton cycle"; I'll edit accordingly. $\endgroup$1more comment