1 1 D ec 2 00 5 Trees , set compositions , and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Chapoton and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). A number of new Hopf algebra structures (neither commutative or cocommutative) and of enveloping algebra structures are introduced and studied in detail. Among other results, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.