On the number of $OE$-trails for a fixed transition system

The existence of $OE$-trail for a plane Eulerian graph had been established earlier and algorithm of its constructing was suggested. This paper is devoted to a question of enumeration of $OE$-trails for a system of transitions induced by a particular $OE$-trail. The upper bound of this estimation does not exceed the double sum of vertices adjacent the outer face and sum of cutvertices degrees. This bound is reachable if a transition system satisfies any $A$-trail. The number of $OE$-trails for an arbitrary chosen transition system is also examined.