Enumeration of rooted nonseparable outerplanar maps (Q908932)

Through a quadratic functional equation for the generating function, it is proved that the number of combinatorially distinct rooted nonseparable outerplanar maps with m edges and root-face valency n is \[ \frac{(m- 1)!(m-2)!}{(n-1)!(n-2)!(m-n)!(m-n+1)!}. \] The number of rooted nonseparable outerplanar maps with m edges happens to be the Catalan number \[ \frac{(2m-2)!}{(m-1)!m!} \] and the author asks for a bijection between the maps above and rooted plane trees with m-1 edges.