  We denote a polygon as a connected component in Cayley tree of order 2
containing certain number of fix vertices. We found an exact formula for a
polygon counting problem for two cases, in which, for the first case the
polygon contain a full connected component of a Cayley tree and for the second
case the polygon contain two fixed vertices. From these formulas, which is in
the form of finite linear combination of Catalan numbers, one can find the
asymptotic estimation for a counting problem.
