  In this paper, by using the Composition-Diamond lemma for non-associative
algebras invented by A. I. Shirshov in 1962, we give Gr\"{o}bner-Shirshov bases
for free Pre-Lie algebras and the universal enveloping non-associative algebra
of an Akivis algebra, respectively. As applications, we show I.P. Shestakov's
result that any Akivis algebra is linear and D. Segal's result that the set of
all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra
$PLie(X)$ generated by the set $X$. For completeness, we give the details of
the proof of Shirshov's Composition-Diamond lemma for non-associative algebras.
