  In this article we introduce the $\mathbb{N}-$decorated plactic monoid of
type $C$, denoted $Pl^\mathbb{N}(C_n)$, via a finite convergent presentation
$ACol$, with generating set $ACol(C_n)$ consisting of admissible columns, and
an element $\epsilon$. By Squier's coherent completion theorem, this
presentation is extended into a coherent presentation by identifying a family
of generating confluences, i.e. generating $3-$cells. Here the generating
$3-$cells are critical branchings on words of length $3$. We adapt the notions
of crystal structure to $ACol(C_n)^\ast$ , and show that the shape of $3-$cells
is preserved by the action of Kashiwara operators. Thus we reduce the study of
the coherent presentation to only describing the generating $3-$cells whose
source is a word of highest weight. We then introduce combinatorial objects
called $C-$trees which parameterize the words of highest weight in
$ACol(C_n)^\ast$. The $C-$trees allow for simplifying calculations with the
insertion algorithm in type $C$, as introduced in by Lecouvey, and we prove
that the generating $3-$cells in $ACol$ are of shape at most $(4,3)$. As a
consequence, we show that the column presentation of $Pl(C_n)$, as introduced
by Hage, has generating $3-$cells of shape at most $(4,3)$. This contrasts the
situation in type $A$, where the $3-$cells in the column presentation of
$Pl(A_n)$ are of shape at most $(3,3)$.
