  Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
$\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must
have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$. In the case $r=1$ we give a
conjectural classification. We then investigate a quantum version of the main
theorem. We conclude with some open problems and a categorification of the
construction. This paper is a full version of the summary arXiv: 0710.3744.
