  Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$
generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and its
quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free
noncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$,
and the quotient bialgebra
$\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha}|~\alpha(g)\ge 2)$ is an Ore
extension of the well-known Fa\`a di Bruno bialgebra. The image of the free
noncommutative Fa\`a di Bruno bialgebra in the quotient
$\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of the
commutative Fa\`a di Bruno bialgebra from the PBW basis point view. If char
$\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of
$\overline{\mathcal{T}}$: $\overline{\mathcal{T}} \twoheadrightarrow
\overline{\mathcal{T}}_{n}\twoheadrightarrow
\overline{\mathcal{T}}_{n}'(p)\twoheadrightarrow
\overline{\mathcal{T}}_{n}(p)\twoheadrightarrow
\overline{\mathcal{T}}_{n}(p;d_{1}) \twoheadrightarrow \ldots
\twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{j},d_{j-1},\ldots,d_{1})
\twoheadrightarrow \ldots \twoheadrightarrow
\overline{\mathcal{T}}_{n}(p;d_{p-2},d_{p-3},\ldots,d_{1})$ with finite
GK-dimensions. Furthermore, we study the homological properties and the
coradical filtrations of those quotient Hopf algebras.
