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A set-operad is a monoid in the category of combinatorial species withrespect to the operation of substitution. From a set-operad, we give here asimple construction of a Hopf algebra that we call {\em the natural Hopfalgebra} of the operad. We obtain a combinatorial formula for its antipode interms of Shr\"oder trees, generalizing the Hayman-Schmitt formula for the Fa\'adi Bruno Hopf algebra. From there we derive more readable formulas for specificoperads. The classical Lagrange inversion formula is obtained in this way fromthe set-operad of pointed sets. We also derive antipodes formulas for thenatural Hopf algebra corresponding to the operads of connected graphs, the NAPoperad, and for its generalization, the set-operad of trees enriched with amonoid. When the set operad is left cancellative, we can construct a family ofposets. The natural Hopf algebra is then obtained as an incidence reduced Hopfalgebra, by taking a suitable equivalence relation over the intervals of thatfamily of posets. We also present a simple combinatorial construction of anepimorphism from the natural Hopf algebra corresponding to the NAP operad, tothe Connes and Kreimer Hopf algebra.

An antipode formula for the natural Hopf algebra of a set operad