Dual graph of a tree

Question

It is stated here that:

For any connected embedded planar graph G define the dual graph G* by drawing a vertex in the middle of each face of G, and connecting the vertices from two adjacent faces by a curve e* through their shared edge e. Note that G**=G.

Given any planar drawing of a tree $G$ is its dual graph simply $K_1$? But then how does one justify $G^{**}=G$?

Answer 1

The dual graph need not be simple. Here’s a badly drawn example of a black tree with $6$ vertices (and therefore $5$ edges and $1$ face) and its red dual: the dual has $1$ vertex, $5$ edges, all of which are loops, and $6$ faces.