  We build a collection of topological Ramsey spaces of trees giving rise to
universal inverse limit structures, extending Zheng's work for the profinite
graph to the setting of Fra\"{\i}ss\'{e} classes of finite ordered binary
relational structures with the Ramsey property. This work is based on the
Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong
subtrees. Based on these topological Ramsey spaces and the work of
Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that
for each such Fra\"{\i}ss\'{e} class, its universal inverse limit structures
has finite big Ramsey degrees under finite Baire-measurable colourings. For
finite ordered graphs, finite ordered $k$-clique free graphs ($k\geq 3$),
finite ordered oriented graphs, and finite ordered tournaments, we characterize
the exact big Ramsey degrees.
