Let (ωn)0 < n be the sequence known as Integer Sequence A047749http://www.research.att.com/ njas/sequences/A047749 In this paper, we show that the integer ωn enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.