  Dyck paths (also balanced brackets and Dyck words) are among the most heavily
studied Catalan families. This paper is a continuation of [2, 3]. In the paper
we enumerate the terms of the OEIS A036991, Dyck numbers, and construct a
concomitant bijection with symmetric Dyck paths. In the case of binary coding
of Dyck paths we work with compact natural numbers after removing leading
zeros. Analysis of binary suffixes, allowed us to obtain a bijection between
arbitrary A036991 terms and symmetric A036991 terms which encode symmetric Dyck
paths. The bijection generates a forest of unary non-intersecting infinite
trees. The root of each bijection tree is an asymmetric term; the other nodes
are symmetrical. There are an infinite number of such trees. The reader is
offered a software package for working with bijection trees.
