Suppose we have a planar graph with vertices $v_0, \ldots, v_n$, where $n$ is even such that there exist a checkerboard coloring of the regions in the complement such that the black regions are $n$ (non-degerate) triangular faces having clockwise orientation.
Prove that the number of oriented rooted spanning trees, say oriented from the root $v_0$, is an odd number. (An oriented spanning tree rooted at $v$ is an acyclic subgraph in which every vertex other than $v$ has indegree $1$.)
In attached example:
I counted $11$ trees, and my strategy was to put the trees in pairs, but it gets complicated like in the example where we have two the edges from $v_1$ to $v_2$.