  The natural Hopf algebra $\mathcal{N} \mathcal{O}$ of an operad $\mathcal{O}$
is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We
introduce new bases of these Hopf algebras deriving from free operads via new
lattice structures on their basis elements. We construct polynomial
realizations of $\mathcal{N} \mathcal{O}$ by using alphabets of variables
endowed with unary and binary relations. By specializing our polynomial
realizations, we discover links between $\mathcal{N} \mathcal{O}$ and the Hopf
algebra of word quasi-symmetric functions of Hivert, the decorated versions of
the Connes-Kreimer Hopf algebra of Foissy, the Fa\`a di Bruno Hopf algebra and
its deformations, and the noncommutative multi-symmetric functions Hopf
algebras of Novelli and Thibon.
