A hierarchy of chains embeddable into the lexicographic power $R^\omega_\lex$

A hierarchy of chains is a transfinite sequence of linear orderings such that each chain in the sequence order-embeds into all chains following it but not in those preceding it. We construct a $c^+$-long hierarchy of chains that order-embed into the lexicographic power $(R^ω,>_\lex)$. Each linear ordering L in this hierarchy is such that there exists a tree representation of L, which is an R-branching tree with no infinite branches. The existence of such a hierarchy sheds some light on the hidden complexity of $(R^ω,>_\lex)$.