  Nichols algebras naturally appear in the classification of finite dimensional
pointed Hopf algebras. Assuming only that the base field has characteristic
zero several new finite dimensional rank 2 Nichols algebras of diagonal type
are listed. Each of them is described in terms of generators and relations. A
Poincar\'e--Birkhoff--Witt basis of all of these Nichols algebras is given and
their dimension is computed. Together with the continuation of this paper this
gives an answer to a question of Andruskiewitsch.
  Key Words: Hopf algebra, Lyndon words, full binary tree
