  In this article, we look into the tree gonality of genus $3$ metric graphs
$\Gamma$ which is defined as the minimum of degrees of all tropical morphisms
from any tropical modification of $\Gamma$ to any metric tree. It is denoted by
tgon$(\Gamma)$ and is at most $3$. We define hyperelliptic metric graphs in
terms of tropical morphisms and tree gonality. Let $\Gamma$ be a genus $3$
metric graph with tgon$(\Gamma) = 3$ which is not hyperelliptic. In this paper,
for such metric graphs $\Gamma$, we construct a tropical modification $\Gamma'$
of $\Gamma$, a metric tree $T$ and a tropical map $\varphi:\Gamma' \to T$ of
degree $3$.
