 Solution

A plane tree has one face of a plane graph, so its dual graph has one vertex. Every original edge is a bridge incident to that same face on both sides, so every dual edge is a loop. Thus the dual drawing in the figure is a bouquet of eight loops, arranged with the displayed nesting inherited from the tree. It has nine faces of a plane graph, matching $f^{\ast }=n=9$. The nesting matters for which faces meet; a bouquet drawn as eight unrelated petals would have a different face-adjacency graph.