  We introduce normalized Drinfeld modular curves that parameterize rank $m$
Drinfeld modules compatible with a $T$-torsion structure arising from a given
conjugacy class of nilpotent upper-triangular $n\times n$ matrices with rank
$\geqslant n-m$ over a finite field $\mathbb{F}_q$. This creates a deep link
connecting the classification of nilpotent upper-triangular matrices and the
decomposition of Drinfeld modular curves. The conjugacy classes of nilpotent
upper-triangular matrices one-to-one corresponds to certain $T$-torsion flags,
and form a tree structure. As a result, the associated Drinfeld modular curves
are organized in the same tree. This generalizes the tower structure introduced
by Bassa, Beelen, Garcia, Stichtenoth, and others. Additionally,we prove the
geometric irreducibility of $(3,2)$-type normalized Drinfeld modular curves,
and characterize their associated function fields.
