  In this paper, we establish a structure theorem for connected graded Hopf
algebras over a field of characteristic $0$ by claiming the existence of a
family of homogeneous generators and a total order on the index set that
satisfy some desirable conditions. The approach to the structure theorem is
constructive, based on the combinatorial properties of Lyndon words and the
standard bracketing on words. As a surprising consequence of the structure
theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov
dimension over a field of characteristic $0$ are all iterated Hopf Ore
extensions of the base field. In addition, some keystone facts of connected
Hopf algebras over a field of characteristic $0$ are observed as corollaries of
the structure theorem, without the assumptions of having finite
Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base
field is algebraically closed.
