  We find a formula to compute the number of the generators, which generate the
$n$-filtered space of Hopf algebra of rooted trees, i.e. the number of
equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of
rooted trees, we show that the analogue of Andruskiewitsch and Schneider's
Conjecture is not true. The Hopf algebra of rooted trees and the enveloping
algebra of the Lie algebra of rooted trees are two important examples of Hopf
algebras. We give their representation and show that they have not any nonzero
integrals. We structure their graded Drinfeld doubles and show that they are
local quasitriangular Hopf algebras.
