  Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the
second named author introduced rooted tree maps as a family of linear maps on
the noncommutative polynomial algebra in two letters. These give a class of
relations among multiple zeta values, which are known to be a subclass of the
so-called linear part of the Kawashima relations. In this paper we show the
opposite implication, that is the linear part of the Kawashima relations is
implied by the relations coming from rooted tree maps.
