Simple Arborification in Hopf Algebra

• Simple arborification is a surjective Hopf algebra morphism that linearizes tree combinatorics by mapping decorated rooted forests to shuffle words.
• It bridges tree-indexed and word-indexed formalisms, playing a key role in multiple zeta values, rough differential equations, and dispersive PDE normal-form analysis.
• Its recursive definition and compatibility with shuffle products enable clear algebraic cancellations and renormalization across various combinatorial and analytic settings.

Searching arXiv for recent and foundational papers on simple arborification and closely related formulations. Simple arborification is a surjective Hopf algebra morphism from a Hopf algebra of decorated rooted forests to a shuffle Hopf algebra of words. In the standard formulation, it maps the combinatorics of rooted trees in the Butcher–Connes–Kreimer framework to ordered words by summing all linearizations compatible with the tree structure, thereby turning branched expansions into shuffle-type word expansions (Manchon, 2016). Across the literature, the notion appears as the shuffle-case counterpart of contracting arborification, and serves as a canonical bridge between tree-indexed and word-indexed formalisms in multiple zeta values, rough differential equations, mould calculus, and dispersive PDE normal-form analysis (Bruned et al., 2018). The term should be distinguished from graph-theoretic arboricity: despite the lexical similarity, simple arborification concerns Hopf-algebraic and combinatorial transformations on rooted forests, not coverings of graph edges by forests (Dai et al., 20 Dec 2025).

1. Definition and core construction

In the shuffle setting, simple arborification is described as a surjective Hopf algebra morphism

$\mathfrak{a} : \mathcal{H}^A_{BCK} \to \mathcal{H}_\shuffle$

from the Butcher–Connes–Kreimer Hopf algebra of $A$-decorated rooted forests to the shuffle Hopf algebra over words on $A$ (Bruned et al., 2018). The same construction is also expressed as

$$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$

with $T(A)$ the tensor or shuffle Hopf algebra of words (Bruned, 2024).

The recursive definition is central. If a decorated tree is written as $B_+^a(\tau_1,\dots,\tau_n)$, with root decoration $a$ and immediate subtrees $\tau_1,\dots,\tau_n$, then

$\mathfrak{a}\big( B_+^a(\tau_1, ..., \tau_n) \big) = \left( \mathfrak{a}(\tau_1) \shuffle \cdots \shuffle \mathfrak{a}(\tau_n) \right) a .$

The empty forest is sent to the empty word, and on forest products the map is multiplicative with respect to the shuffle product (Bruned, 2024). In another equivalent formulation, for a forest $F=\tau_1\cdots\tau_n$,

$A$0

(Bruned et al., 2018).

Operationally, the map sends a decorated forest to the sum of all words obtained by linearizing the partial order induced by the rooted-tree structure. Ladder trees map to single words, whereas genuinely branched trees map to sums of shuffle-compatible words (Manchon, 2016). This is why the process is often described as “linearizing” tree combinatorics into word combinatorics (Fan, 28 Aug 2025).

A more structural characterization uses the universal property of decorated rooted forests. Given a graded Hopf algebra $A$1 and Hochschild one-cocycles $A$2, there exists a unique Hopf algebra morphism $A$3 satisfying

$A$4

For simple arborification, the target cocycle is right concatenation in the shuffle algebra (Manchon, 2016). This universal-property viewpoint is one of the main reasons the construction appears in several mathematically distinct areas.

2. Hopf-algebraic framework and formal properties

Simple arborification is defined relative to two Hopf algebras. On the tree side stands the Butcher–Connes–Kreimer Hopf algebra of decorated rooted forests, with commutative forest product and coproduct given by admissible cuts. On the word side stands the shuffle Hopf algebra, with shuffle product and deconcatenation coproduct (Bruned, 2024).

Its defining algebraic property is compatibility with both products and coproducts. In particular,

$A$5

and

$A$6

so $A$7 is a Hopf algebra morphism (Bruned, 2024). Surjectivity holds because every word is the image of a ladder tree (Bruned, 2024).

Several papers emphasize the contrast with contracting arborification. In the shuffle case, no contractions of letters occur; only shuffling and concatenation appear. In the quasi-shuffle case, contractions are allowed according to a semigroup product on the alphabet, yielding the map usually denoted $A$8 or the contracting arborification (Bruned et al., 2018). This distinction is standard: simple arborification is “the shuffle case,” contracting arborification is “the quasi-shuffle case” (Clavier, 2018).

Alternative recursive formulas clarify the internal structure of the map. One such formula is

$A$9

where $A$0 projects to single-node trees and $A$1 is the adjoint of grafting (Bruned, 2024). A related identity equates a recursion based on the BCK coproduct with one based on adjoint grafting, which suggests that arborification may be viewed either as iterative pruning via cuts or as recursive extraction of letters via grafting duality (Bruned, 2024).

The older mould-calculus literature situates arborification in a universal Hopf-algebraic setting. There it appears as a factorization of characters involving shuffle or quasi-shuffle Hopf algebras and the Connes–Kreimer Hopf algebra, with the latter enjoying a universal property that guarantees the existence of the transformation (Fauvet et al., 2012). This suggests that simple arborification is not an ad hoc combinatorial device, but part of a broader categorical pattern governing recursive algebraic expansions.

3. Relations to contracting arborification and allied transforms

The most important neighboring notion is contracting arborification. In rough-path and stochastic contexts, one distinguishes

$A$2

the latter landing in a quasi-shuffle Hopf algebra (Bruned et al., 2018). The simple map encodes linear extensions only; the contracting map also permits mergers of letters dictated by the semigroup structure.

This distinction is explicit in the multiple-zeta literature. In the integral picture, forests are decorated by two colors and mapped by simple arborification to shuffle words. In the series picture, forests are decorated by positive integers and mapped by contracting arborification to quasi-shuffle words (Manchon, 2016). The difference is therefore not merely algebraic but tied to two analytic realizations: iterated integrals versus iterated sums.

A further development concerns maps between the two arborified worlds. For arborified multiple zeta values, one has simple arborification

$A$3

for $A$4 and contracting arborification

$A$5

for $A$6 (Fan, 28 Aug 2025). The paper constructs, for planar rooted trees, a map between the two BCK Hopf algebras making the natural diagram with the word-level morphism $A$7 commute (Fan, 28 Aug 2025). This resolves a problem attributed there to Manchon. The result is significant because a naive ladder-tree section preserves the word-level diagram but destroys much of the tree structure; the planar recursive construction preserves more of the arborified combinatorics (Fan, 28 Aug 2025).

Another neighboring transform is the flattening map $A$8, used in the theory of arborified zeta values to pass from trees to words. In that setting, simple arborification is the general procedure of lifting word-level structures to trees and then flattening back to words as needed (Clavier, 2018). This suggests an editorially useful distinction: simple arborification is often the structural principle, while flattening is the explicit tree-to-word morphism in a particular operated-algebra construction.

4. Multiple zeta values and arborified zeta values

Simple arborification plays a central role in the theory of arborified multiple zeta values. In the integral picture of multiple zeta values, decorated rooted forests with two colors are sent to shuffle words, and the resulting objects are interpreted through iterated integrals (Manchon, 2016). The map is again a surjective Hopf algebra morphism, and the image of a forest is the sum over all total orders compatible with the forest partial order (Manchon, 2016).

Examples in this literature illustrate the combinatorial effect of branching. A tree may map to a linear combination such as

$A$9

which then yields the corresponding linear combination of multiple zeta values under the zeta character (Manchon, 2016). Ladder trees produce a single word, while branched trees produce multiple shuffle words with combinatorial coefficients.

The theory of arborified zeta values extends these constructions to tree-level products. Generalizations of shuffle and stuffle products are defined directly on rooted trees, and arborified zeta value maps become algebra morphisms for the corresponding products on trees (Clavier, 2018). In that framework, simple arborification underlies the shuffle side, while contracting arborification underlies the stuffle side. The paper shows that arborified zeta values are finite $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$0-linear, indeed $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$1-linear, combinations of convergent multiple zeta values (Clavier, 2018).

A recurrent theme is that tree branching introduces relations absent at the word level. Because the generalized tree shuffle is commutative, non-associative, and unital, its associators yield new relations whose images under arborified zeta values vanish (Clavier, 2018). This indicates that simple arborification is not merely a repackaging of multiple zeta values: it enriches the algebraic setting by retaining non-ladder tree structure before passage to words.

The later planar-tree work on a map between the two arborifications of multiple zeta values makes this point sharper. The problem there is precisely to relate the integral-side simple arborification and the series-side contracting arborification at the tree level rather than only at the word level (Fan, 28 Aug 2025). A plausible implication is that the tree-level viewpoint captures compatibility phenomena obscured by direct word-level translation.

5. Dispersive PDEs and normal-form expansions

In recent work on dispersive PDEs, simple arborification appears as a mechanism for turning decorated trees into words so that cancellations become algebraically transparent. One paper studies cancellations for dispersive PDEs with random initial data and defines an arborification map $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$2 sending decorated trees arising from iterated Duhamel expansions to sums of words, with shuffle products encoding all possible interleavings compatible with tree partial orders (Bruned et al., 2024).

The recursive formulation matches the standard simple-arborification pattern: apply $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$3 to subtrees, shuffle the results, then concatenate the root letter (Bruned et al., 2024). The key analytic point is that once tree expansions are arborified, cancellations that were difficult to see among iterated integrals become algebraic cancellations among words. The paper presents this as an alternative to the molecule formalism introduced by Deng and Hani (Bruned et al., 2024).

A closely related work on derivation of normal forms for dispersive PDEs via arborification states explicitly that the key tool is the arborification map from the Butcher–Connes–Kreimer Hopf algebra to the Shuffle Hopf algebra (Bruned, 2024). There the transformation systematizes normal-form derivations using decorated trees, with Hopf-algebraic identities organizing decomposition formulas. The same paper records the standard recursive definition and the compatibility with coproducts and products (Bruned, 2024).

The later work on resonance-based schemes via normal forms continues this line. It uses an arborification map on decorated trees together with a Butcher–Connes–Kreimer type coproduct and lower-dominant-parts decompositions to derive a family of low regularity schemes with explicit formulae for coefficients and local error (Bruned, 11 Nov 2025). The map

$$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$4

is again the organizing device that converts tree-indexed expansions into word-indexed expressions used in normal-form reductions (Bruned, 11 Nov 2025).

These PDE applications should not be mistaken for independent reinventions. They use the same shuffle-side arborification pattern that appears in earlier algebraic work. What changes is the semantic content of the letters and trees: instead of representing iterated sums or rough-path signatures, they encode oscillatory phases, nonlinear interactions, pairings, or integration-by-parts histories (Bruned et al., 2024).

6. Rough paths, renormalisation, and mould calculus

In rough differential equations, simple arborification is one half of a broader comparison between shuffle and quasi-shuffle worlds. The paper on quasi-shuffle algebras and renormalisation of rough differential equations describes arborification as a method for relating the combinatorics of rooted trees to words and their associated shuffle or quasi-shuffle Hopf algebras (Bruned et al., 2018). There, the simple case is the shuffle morphism $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$5, while the quasi-shuffle case is the contracting morphism $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$6.

A central application is the arborified Hoffman exponential, which establishes a Hopf algebra isomorphism between shuffle and quasi-shuffle structures and yields a canonical renormalisation coinciding with Marcus’ canonical extension for semimartingale driving signals (Bruned et al., 2018). In this context, simple arborification is part of an algebraic infrastructure for transferring problems from branched rough paths to word-based Hopf algebras where quasi-shuffle automorphisms are more tractable.

The paper on a comodule-bialgebra structure for word-series substitution and mould composition places the construction in Ecalle’s mould-calculus environment. It introduces an internal coproduct on the quasi-shuffle Hopf algebra compatible with the quasi-shuffle product and shows that arborification intertwines the internal coproducts on forests and words (Ebrahimi-Fard et al., 2016). The paper states that the same formalism applies to simple arborification by replacing quasi-shuffle with shuffle (Ebrahimi-Fard et al., 2016).

The earlier paper on Ecalle’s arborification-coarborification transforms frames the matter even more broadly. Arborification is presented as a Hopf-algebraic morphism arising from universal properties of the Connes–Kreimer algebra, while coarborification is the dual process sending arborescent objects to differential operators (Fauvet et al., 2012). In this literature, the conceptual role of simple arborification is to replace sequence-indexed expansions by tree-indexed ones in a way that improves structural clarity and, in dynamical-systems applications, analytic control (Fauvet et al., 2012).

This suggests a general interpretation: simple arborification is a normalization of dependency structure. Word-based expansions privilege total order; tree-based expansions retain branching; the arborification map mediates between them without losing the shuffle algebra governing ordered iterated operations.

7. Scope, terminology, and common confusions

The phrase “simple arborification” has a specific technical meaning in the cited literature. It denotes the shuffle-based arborification map from decorated rooted forests to shuffle words. It does not denote Matula’s number-theoretic correspondence between integers and rooted trees, even though that correspondence is also called “arborification” in another sense (Manchon, 3 Feb 2026). Matula’s construction associates each positive integer with a rooted tree or forest reflecting prime factorization; it is bijective and number-theoretic, whereas simple arborification in the Ecalle–BCK–shuffle tradition is a Hopf-algebraic morphism between combinatorial algebras (Manchon, 3 Feb 2026).

The term should also be separated from graph arboricity and related algorithmic notions such as arboricity approximation or the arboricity polynomial. Those topics concern the minimum number of forests needed to cover edges of a graph, sublinear algorithms for estimating that number, constructive decompositions of graphs into forests, or counting labeled independent-set covers in matroids and graphs (Blumenstock et al., 2018). They are mathematically unrelated to simple arborification except for the shared lexical root “arbor-”.

Within the algebraic literature itself, the main internal distinction is between simple and contracting arborification. The former uses the shuffle product and no contractions; the latter uses quasi-shuffle-type contractions and corresponds naturally to sum-side or Itô-type structures (Bruned et al., 2018). Confusing the two obscures the algebraic meaning of the target Hopf algebra and the analytic interpretation of the resulting words.

A final source of ambiguity concerns whether arborification is being used as a tree-to-word map or more broadly as a strategy for translating between branched and ordered expansions. The papers support both usages. In strict formal terms, simple arborification is the morphism $$\mathfrak{a}: \mathcal{H}^A_{\mathrm{BCK}} \longrightarrow T(A),$$7 itself (Bruned, 2024). In a broader methodological sense, it is the principle that tree-indexed recursive objects can be reorganized through shuffle-compatible word expansions, often revealing cancellations, universal constructions, or renormalisation mechanisms (Bruned et al., 2024).