Arborified Multiple Zeta Values

• Arborified multiple zeta values are generalized MZVs that replace linear word orders with partially ordered rooted trees, offering a flexible combinatorial structure.
• They introduce tree-level analogues of shuffle and quasi-shuffle products via Hopf algebra morphisms, recursive corrections, and planar tree techniques.
• Convergent arborified values reduce to finite Q-linear combinations of classical MZVs, with explicit series representations and dendriform algebra frameworks underlying their structure.

Arborified multiple zeta values are rooted-tree and rooted-forest generalizations of classical multiple zeta values in which the linear order of an index or word is replaced by the partial order of a rooted tree. In Écalle’s arborification formalism, the shuffle and quasi-shuffle structures of ordinary multiple zeta values are lifted from words to decorated rooted forests through Hopf algebra morphisms, producing tree-indexed zeta quantities in both a series picture and an iterated-integral picture (Manchon, 2016). Subsequent work established explicit series representations, tree-level analogues of shuffle and stuffle structures, reductions to classical multiple zeta values, links with Mordell–Tornheim, conical, and Shintani zeta values, and a natural map reconciling the two principal arborification procedures at the tree level (Clavier et al., 2022, Clavier, 2018, Catoire et al., 27 Aug 2025, Fan, 28 Aug 2025).

1. Classical origin and the passage from words to trees

Classical multiple zeta values admit both nested-sum and iterated-integral realizations. One standard sum formula is

$$\zeta(n_1,\ldots,n_r) = \sum_{k_1>k_2>\cdots>k_r\ge 1} \frac{1}{k_1^{n_1}\cdots k_r^{n_r}},$$

convergent when $n_1\ge 2$, while one iterated-integral formula is

$$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$

with

$$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$

These two realizations underlie the quasi-shuffle relations from nested sums and the shuffle relations from iterated integrals (Manchon, 2016, Fan, 28 Aug 2025).

Arborified multiple zeta values replace the chain structure of a word by a rooted tree or rooted forest. The tree encodes a partial order, and the summation or integration variables are constrained by that order rather than by a total order. In this sense, ordinary multiple zeta values correspond to the ladder-tree sector, while branching vertices introduce genuinely new combinatorics. The 2016 formulation already emphasized that words correspond to ladders and that trees encode “all ways of nesting or interleaving more flexibly than words” (Manchon, 2016).

This replacement of words by trees is not merely notational. It produces a setting in which the two classical realizations of multiple zeta values can be studied simultaneously at the level of rooted-tree combinatorics. A recurring theme is that branching creates new coefficients, new product structures, and new comparison problems, but does not introduce a new class of transcendental constants: arborified values reduce to finite combinations of ordinary multiple zeta values (Clavier, 2018, Clavier et al., 2022).

2. Hopf algebras, arborification morphisms, and word-level comparison

The basic combinatorial object is the Butcher–Connes–Kreimer Hopf algebra $H_{BCK}^D$ of $D$-decorated rooted trees, with grafting operators $B_+^d$ and the standard pruning/cut coproduct. The universal property used in arborification is the following: if $\mathcal H$ is a graded Hopf algebra and $L^d$ is a Hochschild $1$-cocycle for each decoration $n_1\ge 2$0, then there exists a unique Hopf algebra morphism

$n_1\ge 2$1

such that

$n_1\ge 2$2

This mechanism is the abstract source of both simple and contracting arborification (Manchon, 2016).

On the integral side one uses the non-commutative polynomial algebra $n_1\ge 2$3 generated by $n_1\ge 2$4 or, in another notation, $n_1\ge 2$5, endowed with the shuffle Hopf structure. On the series side one uses $n_1\ge 2$6, generated by $n_1\ge 2$7, with its stuffle or quasi-shuffle Hopf structure. Manchon’s two principal arborification morphisms are

$n_1\ge 2$8

characterized by

$n_1\ge 2$9

where $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$0 is right concatenation. In the earlier notation these are $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$1 and $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$2; the former is the simple arborification and the latter the contracting arborification (Fan, 28 Aug 2025, Manchon, 2016).

At the word level, the sum and integral pictures are related by the standard encoding

$$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$3

or equivalently, in the $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$4-notation,

$$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$5

For convergent words this gives the classical identification of the two realizations. After regularization, however, the two extensions differ by the Boutet de Monvel–Zagier operator

$$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$6

which already indicates that a tree-level comparison is structurally delicate (Manchon, 2016).

3. Series-side and integral-side arborified zeta values

In the series picture, arborified multiple zeta values are attached to $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$7-decorated rooted trees. If $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$8 is a rooted tree with opposite partial order $$\zeta(k_1,\dots,k_d) = (-1)^d\, I(0;1,0^{k_1-1},\dots,1,0^{k_d-1};1),$$9 and vertex decorations $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$0, then

$$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$1

This is the rooted-tree analogue of the harmonic-series definition. In the formulations appearing in the literature, convergence is expressed through the decorations governing the extremal vertices of the order: one presentation states that the series converges when the root is decorated by $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$2 with $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$3, another defines convergent trees as those whose root decoration is $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$4, and the 2016 contracted sum picture records convergence when every leaf has decoration $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$5 (Fan, 28 Aug 2025, Clavier, 2018, Manchon, 2016).

In the integral picture, arborified multiple zeta values are attached to $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$6-decorated rooted trees, with $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$7 or $$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$8. One definition is

$$I(a_0;a_1,\dots,a_k;a_{k+1}) = \int_{a_0<t_1<\cdots<t_k<a_{k+1}} \prod_{j=1}^k \frac{dt_j}{t_j-a_j}.$$9

where

$H_{BCK}^D$0

and

$H_{BCK}^D$1

Equivalent formulations in the $H_{BCK}^D$2-notation use

$H_{BCK}^D$3

The corresponding convergence conditions are presented as root labeled $H_{BCK}^D$4 and leaves labeled $H_{BCK}^D$5, or roots decorated by $H_{BCK}^D$6 and leaves and branching vertices by $H_{BCK}^D$7, depending on the chosen conventions (Fan, 28 Aug 2025, Clavier et al., 2022).

These two definitions are the tree analogues of the classical sum and iterated-integral pictures. The series side uses positive-integer decorations and quasi-shuffle combinatorics; the integral side uses two-color decorations and shuffle combinatorics. In both cases the rooted-tree partial order governs the domain of summation or integration, so ladder trees recover the word case while branched trees interpolate genuinely non-linear order structures (Manchon, 2016, Fan, 28 Aug 2025).

4. Reduction to classical multiple zeta values and related zeta families

A central structural fact is that arborified zeta values are finite linear combinations of ordinary multiple zeta values. This follows from flattening or arborification morphisms that send a forest to a finite sum of words, after which the usual zeta map is applied. In the 2018 framework, for any convergent forest $H_{BCK}^D$8, the stuffle-side and shuffle-side arborified zeta values are finite $H_{BCK}^D$9-linear combinations of classical multiple zeta values, in fact with integer coefficients (Clavier, 2018). In the 2022 framework, for convergent $D$0-decorated forests,

$D$1

hence every convergent arborified zeta value is a finite $D$2-linear combination of multiple zeta values (Clavier et al., 2022).

The 2022 paper also established an explicit series representation for every convergent arborified zeta value. If $D$3 is a convergent $D$4-decorated rooted forest and $D$5 is its set of segments, then

$D$6

This theorem motivates the introduction of tree zeta values

$D$7

and the branched binarisation map $D$8, for which

$D$9

on convergent $B_+^d$0-decorated forests (Clavier et al., 2022).

This reduction principle extends beyond ordinary arborified values. The same 2022 analysis shows that convergent tree zeta values are $B_+^d$1-linear combinations of multiple zeta values, derives explicit formulas for Mordell–Tornheim zeta values, proves that convergent arborified zeta values are conical zeta values, and characterizes which conical zeta values are tree-like (Clavier et al., 2022). A common misconception is therefore excluded by the available results: arborified zeta values enlarge the combinatorial indexing system from words to rooted forests, but the values themselves remain within the multiple-zeta-value algebraic span.

5. Tree-level shuffle, stuffle, and finer algebraic structures

One major development is the construction of genuine products on rooted forests that generalize the classical shuffle, stuffle, and anti-stuffle products. In the 2018 treatment, for $B_+^d$2 one has tree-level $B_+^d$3-shuffle products $B_+^d$4 on forests, with $B_+^d$5 giving the stuffle product on trees, $B_+^d$6 the shuffle product on trees, and $B_+^d$7 the anti-stuffle product on trees. These algebras are commutative and unital but generally nonassociative. Arborified zeta maps are algebra morphisms for these products, and the associators lie in the kernels of the corresponding zeta maps, producing relations with no direct word-level analogue (Clavier, 2018).

This tree-level multiplicativity is tied to Rota–Baxter theory. The same work shows that a linear map $B_+^d$8 on a commutative algebra is a Rota–Baxter operator of weight $B_+^d$9 if and only if it is an algebra morphism for the $\mathcal H$0-shuffle product on trees. The result places arborified zeta values in the same structural orbit as classical multiple zeta values, but now with branching-dependent product identities (Clavier, 2018).

A later refinement replaces these generally nonassociative rooted-forest products by associative products arising from tridendriform and dendriform structures on Schröder trees and binary trees. The 2025 paper on Schröder trees shows that quasi-shuffle decomposes into tridendriform operations $\mathcal H$1, while shuffle decomposes into dendriform operations $\mathcal H$2. Using the free tridendriform and dendriform algebra structures on decorated Schröder trees and decorated binary trees, it defines tridendriform zeta values and dendriform zeta values as universal morphisms into spaces of formal series and formal integrals. After evaluation, these recover arborified zeta values and imply that arborified zeta values are algebra morphisms for associative quasi-shuffle and shuffle products on trees (Catoire et al., 27 Aug 2025).

This later perspective suggests a refinement of the earlier double-shuffle picture. Rather than treating shuffle and quasi-shuffle as monolithic associative products, one views them as shadows of finer dendriform or tridendriform structures. At the evaluated level, the same paper relates the integral dendriform construction to Shintani zeta values by showing that every evaluated dendriform zeta value can be expressed as a Shintani zeta value of explicitly constructed data $\mathcal H$3 (Catoire et al., 27 Aug 2025).

6. Comparison of the two arborifications and the planar-tree solution

A longstanding problem in the subject is to lift the classical word-level comparison map from $\mathcal H$4 to $\mathcal H$5 to a natural map between the two Hopf algebras of decorated rooted trees. Manchon asked for a tree map

$\mathcal H$6

such that

$\mathcal H$7

Earlier work had already emphasized that the obvious choice collapses trees into ladders and destroys the geometry of the forest structure, so a genuinely tree-respecting comparison was missing (Manchon, 2016).

The 2025 paper “A map between arborifications of multiple zeta values” resolves this compatibility problem by passing first to planar rooted trees. It introduces the non-commutative Hopf algebra $\mathcal H$8 of $\mathcal H$9-decorated planar rooted trees, with forgetful projections

$L^d$0

and lifted arborification maps

$L^d$1

The construction then exploits a recursive decomposition of non-ladder planar rooted trees by means of a minimal incomparable pair $L^d$2, introduces an error term $L^d$3 measuring the failure of naive compatibility, and packages the recursive corrections into a process tree $L^d$4. From this data the paper defines

$L^d$5

and the corrected planar map

$L^d$6

Its main planar theorem is the commutative diagram

$L^d$7

Finally, averaging over all planar structures gives a section

$L^d$8

and the desired non-planar comparison map

$L^d$9

which satisfies

$1$0

This supplies the sought tree-level bridge between the series and integral arborifications (Fan, 28 Aug 2025).

The same paper clarifies why naive binarisation was insufficient. Clavier’s natural map $1$1 on trees does not make the comparison diagram commute; instead it yields

$1$2

with equality only for ladder forests. The planar recursive correction therefore isolates branching as the precise source of the mismatch between the two arborified pictures. This suggests that the compatibility of sum-side and integral-side arborification is not visible at the level of bare non-planar trees alone, but emerges after keeping track of planar structure and then descending back to the ordinary rooted-tree setting (Fan, 28 Aug 2025).