  We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra
of non-commutative symmetric functions described as a subalgebra of the of
rooted ordered coloured trees Hopf algebra. Its natural basis can be identified
with the set of sentences over alphabet $A$ (the set of colours). We present
also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric
functions together with its realization in terms of power series in partially
commutative variables. We provide formulas expressing multiplication,
comultiplication and the antipode for these Hopf algebras in various bases --
the corresponding generalizations of the complete homogeneous, elementary,
ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and
fundamental bases of $\mathrm{QSym}$. We study also certain distinguished
series of trees in the setting of restricted duals to Hopf algebras.
