  A graph is closed when its vertices have a labeling by $[n]$ such that the
binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the
lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In
this paper, we generalize this notion and study the so called $m-$closed
graphs. We find equivalent condition to $3-$closed property of an arbitrary
tree $T$. Using it, we classify a class of $3-$closed trees. The primary
decomposition of this class of graphs is also studied.
