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The left Butcher product s ∘ t of two rooted trees s and t is defined by grafting s on the root of t. For example:

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The following theorem is due to Dzhumadil’daev and Löfwall (2002) (see Livernet (2006) for an operadic approach):

THEOREM 1.5.– The free NAP algebra with d generators is the vector space spanned by rooted trees with d colors, endowed with the left Butcher product.

PROOF.– We give the proof for one generator, the case of d generators being entirely similar. The left NAP property for the Butcher product is obvious. Let (A, ▶) be any left NAP algebra, and let a ∈ A. We have to prove that there exists a unique left NAP algebra morphism Ga from to (A, ▶), such that Ga(•) = a. As in the pre-Lie case, we proceed by double induction, first on the number n of vertices, and second on the number k of branches. In the case k = 1, the tree t writes B₊(t₁) = t₁ ∘ ∙; hence, Ga(t) = Ga(s) ▶ a is the only possible choice. Now a tree with k branches writes:

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The only possible choice is then:

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and the result is clearly symmetric in t₁ and t₂ due to the left NAP identity in A. Using the induction hypothesis, the result is also invariant under permutation of the branches 2,3,…,k. Hence, it is invariant under the permutation of all branches, which proves the theorem. □

Despite the similarity with the pre-Lie situation described in section 1.6.2, the NAP framework is much simpler due to the set-theoretic nature of the Butcher product: for any trees s and t, the Butcher product s ∘ t is a tree, whereas the grafting s → t is a sum of trees. We obtain for the first trees: