---
title: Arborified Zeta Values in Rooted Forests
url: https://www.emergentmind.com/topics/arborified-zeta-values
type: topic
---

# Arborified Zeta Values in Rooted Forests

Searching arXiv for recent and foundational papers on arborified zeta values and closely related generalizations.
Arborified zeta values are rooted-forest analogues of multiple zeta values in which the linear combinatorics of words is replaced by the branching combinatorics of decorated rooted trees and forests. They admit both iterated-series and iterated-integral realizations, are constructed through the universal properties of rooted forests and the Butcher–Connes–Kreimer framework, and reduce in the convergent setting to finite $\mathbb Q$-linear combinations of classical multiple zeta values. Their distinctive content lies not in producing new convergent periods but in importing genuinely tree-level phenomena—branching, contractions, nonassociative products, and new double-shuffle-type relations—into zeta-value theory [1812.00777][1603.01498].

## 1. Rooted forests as indices for zeta values

The foundational move is to replace words by rooted forests. A rooted tree is a finite connected loopless graph with a distinguished root and an induced partial order on vertices; a rooted forest is a finite disjoint union of rooted trees. In the relevant constructions, each vertex is decorated by an element of a set such as $\mathbb N^*$, $Y=\{y_n\mid n\ge 1\}$, or $\{x,y\}$, depending on whether one is working on the series or integral side [1812.00777][1603.01498].

This replacement preserves the two classical realizations of multiple zeta values. On the word side, multiple zeta values appear both as nested sums
$$
\zeta(n_1,\dots,n_k)=\sum_{m_1>\cdots>m_k>0}\frac{1}{m_1^{n_1}\cdots m_k^{n_k}}
$$
and as Chen iterated integrals with
$$
\omega_x=\frac{dt}{t},\qquad \omega_y=\frac{dt}{1-t}.
$$
Arborified zeta values keep these two pictures but index them by rooted forests rather than words. Ladder trees recover the word case, while branching allows several incomparable branches to interact combinatorially in a way unavailable for words [1812.00777].

In Manchon’s formulation, a rooted tree decorated by positive integers yields a tree-indexed finite sum in which the inequalities among summation variables are dictated by the tree order, and a two-coloured rooted tree yields an iterated integral over a simplex cut out by the same partial order. The multiplicative extension to forests is part of the basic setup, so forests play the role of commutative products of connected tree components [1603.01498].

## 2. Universal properties, grafting, and arborification

A central structural tool is the branching operator
$$
B_+^\omega(F),
$$
which grafts a new root labelled by $\omega$ on top of the roots of a forest $F$. The initiality theorem for rooted forests states that for any commutative $\Omega$-operated algebra $(A,B)$ there exists a unique algebra morphism
$$
\phi:F_\Omega\to A
$$
such that
$$
\phi(B_+^\omega(F))=B^\omega(\phi(F)).
$$
This universal property is the mechanism that lifts label-level operators to forest-level constructions and underlies the recursive definition of arborified zeta values [1812.00777].

The same idea is expressed Hopf-algebraically through the Butcher–Connes–Kreimer Hopf algebra of decorated rooted forests. In that setting, forests form the free commutative unital algebra on decorated rooted trees, the coproduct is defined by admissible cuts, and the grafting operators $B_+^d$ are Hochschild $1$-cocycles. A general theorem then yields, from any suitable family of Hochschild cocycles $L^d$, a unique Hopf algebra morphism $\Phi_L$ satisfying
$$
\Phi_L\circ B_+^d=L^d\circ\Phi_L.
$$
This is the formal source of arborification maps [1603.01498][2508.20387].

Two such morphisms are fundamental. The simple arborification maps $X$-decorated rooted trees to the shuffle Hopf algebra on $\langle X\rangle=\mathbb Q\langle x_0,x_1\rangle$, while the contracting arborification maps $Y$-decorated rooted trees to the quasi-shuffle Hopf algebra on $\langle Y\rangle=\mathbb Q\langle y_1,y_2,\dots\rangle$. The contracting case is surjective because a word is the image of the corresponding ladder tree. This makes arborification a surjective Hopf algebra morphism from decorated rooted forests onto shuffle or quasi-shuffle word algebras [1603.01498].

## 3. Series and integral realizations

On the series side, the 2018 construction uses log-polyhomogeneous symbols and the Euler–MacLaurin summation operator. For $a\in\mathbb N^*$ one sets
$$
R(a)(x)=x^{-a},
$$
lifts this to forests, and composes with an iterated summation operator built from Euler–MacLaurin to define
$$
\zeta^\#:=\operatorname{ev}_\infty\circ G\circ R^\#.
$$
For a decorated forest $F$, the associated quantity $Z_\lambda(F)(x)$ is an iterated nested sum, and the stuffle arborified zeta value is obtained by the limit
$$
\zeta^\#(F)=\lim_{x\to\infty} Z_\lambda(F)(x),\qquad \lambda\in\{0,-1\}.
$$
The control of asymptotics rests on the filtration properties of the operator $P$ on symbol classes [1812.00777].

On the integral side, arborified Chen integrals are defined recursively by the same branching mechanism. Specializing to
$$
\omega_x(t)=\frac1t,\qquad \omega_y(t)=\frac1{1-t},
$$
one obtains arborified polylogarithms $\operatorname{Li}_F(z)$ for semiconvergent forests. The limit
$$
\zeta^{\amalg}(F)=\lim_{z\to 1}\operatorname{Li}_F(z)
$$
defines the shuffle arborified zeta value for convergent forests in the $\{x,y\}$-decorated setting [1812.00777].

A later refinement shows that every convergent shuffle arborified zeta value admits an explicit series representation. For a convergent $\{x,y\}$-decorated rooted forest $F$, one introduces segments ending at $y$-decorated vertices and proves
$$
\zeta^T(F) = \sum_{\substack{n_v\ge 1\\ v\in V_y(F)}} \prod_{v\in V_y(F)} \left( \sum_{\substack{v'\in V_y(F)\\ v'\succeq v}} n_{v'} \right)^{-|s_v|}.
$$
This shows that the integral definition already has a genuine rooted-forest series expansion, rather than merely an indirect relation to word-indexed iterated integrals [2202.00611].

## 4. Convergence and reduction to classical multiple zeta values

The convergence theory is tree-sensitive but clean. In the $\mathbb N^*$-decorated setting, a decorated rooted tree is convergent if it is empty or if its root decoration satisfies $d(\mathrm{root})\ge 2$, and a forest is convergent if each connected component is convergent. For any convergent forest $F$ and $\lambda\in\{0,-1\}$, the limit $\zeta_\lambda(F)=\lim_{x\to\infty}Z_\lambda(F)(x)$ exists and is finite [1812.00777].

In the integral $\{x,y\}$-decorated setting used for rooted forests, convergence is formulated differently: all roots are decorated by $x$, and all leaves and branching vertices are decorated by $y$. This condition is tailored to the singular forms $dt/t$ and $dt/(1-t)$ and is the analogue of the usual admissibility condition for iterated integrals of multiple zeta type [2202.00611].

A decisive structural fact is that convergent arborified zeta values are not new numbers in the convergent setting. The flattening map
$$
\mathrm{fl}_\lambda:F_\Omega\to W_\Omega
$$
is defined recursively by
$$
\mathrm{fl}_\lambda(\emptyset)=\emptyset,\qquad
\mathrm{fl}_\lambda(F_1F_2)=\mathrm{fl}_\lambda(F_1)\shuffle_\lambda \mathrm{fl}_\lambda(F_2),\qquad
\mathrm{fl}_\lambda(B_+^\omega(F))=\omega\,\mathrm{fl}_\lambda(F),
$$
and Theorem 3.25 states that for any convergent forest $F$, the convergent arborified zeta value $\zeta^\#(F)$ is a finite $\mathbb Q$-linear combination of convergent multiple zeta values, more precisely a finite linear combination with integer coefficients [1812.00777].

The same picture already appears in Manchon’s arborification formalism. Contracted arborified multiple zeta values on positive-integer-decorated trees and simple arborified multiple zeta values on two-colour trees are obtained by composing classical zeta characters with the arborification morphisms, and branching expands a single tree into finitely many ordinary multiple zeta values by allowing permutations of incomparable branches and, in the quasi-shuffle case, mergers of indices [1603.01498].

This resolves a common misconception. Arborified zeta values do not merely rename multiple zeta values, because branching carries new algebraic information; but they also do not, in the convergent regime described above, generate new transcendental numbers beyond finite $\mathbb Q$-linear combinations of classical multiple zeta values. What is new is the forest combinatorics and the relations it induces [1812.00777].

## 5. Tree shuffle, tree stuffle, and arborified double shuffle relations

A central contribution of the 2018 theory is the definition of genuine tree analogues of shuffle, stuffle, and anti-stuffle products. For forests $F,F'$, the recursive product $\shuffle_\lambda$ is characterized by the unit relation
$$
\emptyset\shuffle_\lambda F = F\shuffle_\lambda \emptyset = F,
$$
distribution over concatenation, and, for trees $F=B_+^\omega(f)$ and $F'=B_+^{\omega'}(f')$,
$$
F\shuffle_\lambda F' =
B_+^\omega(f\shuffle_\lambda F')
+ B_+^{\omega'}(F\shuffle_\lambda f')
+ \lambda\, B_+^{\omega\cdot\omega'}(f\shuffle_\lambda f').
$$
The cases $\lambda=1$, $\lambda=0$, and $\lambda=-1$ give the stuffle, shuffle, and anti-stuffle products on trees respectively [1812.00777].

These products are not formal copies of the word operations. For $\lambda\neq 0$ and a commutative semigroup of labels, $(F,\shuffle_\lambda,\emptyset)$ is a commutative, unital, nonassociative algebra. The nonassociativity is intrinsic and reflects genuine branching. In particular, the tree setting has associators
$$
(F_1\shuffle_\lambda F_2)\shuffle_\lambda F_3 - F_1\shuffle_\lambda(F_2\shuffle_\lambda F_3)
$$
that do not disappear algebraically, even though they vanish after evaluation by the corresponding arborified zeta maps [1812.00777].

The main morphism theorem states that
$$
\zeta^\#,\qquad \zeta^\star,\qquad \zeta^{\amalg}
$$
are algebra morphisms for the stuffle, anti-stuffle, and shuffle products on convergent forests. Consequently, the kernels of these maps contain associators, and Corollary 5.13 identifies these vanishing associators as new tree-level relations. This is the arborified form of the double shuffle philosophy: the familiar word-level shuffle and stuffle relations survive, but branching introduces additional relations with no direct word analogue [1812.00777].

An appendix-level comparison with branched binarization sharpens the distinction between ladders and genuinely branched forests. The map
$$
s_\mathcal T:F_{\mathbb N^*}\to F_{\{x,y\}}
$$
extends Hoffman’s binarization to trees, maps convergent forests to convergent forests, and is bijective on that class. The comparison theorem gives
$$
\zeta^\amalg(s_\mathcal T(F))\le \zeta(F),
$$
with equality if and only if $F$ has no branching vertex, i.e. $F$ is a ladder forest. This isolates branching as the source of a strict difference between the tree and word cases [1812.00777].

## 6. Generalizations, comparison maps, and later developments

The rooted-forest framework has been extended in several directions. One major development introduces tree zeta values for $\mathbb N^*$-decorated rooted forests:
$$
\zeta^t(F) := \sum_{\substack{n_v\ge1\\ v\in V(F)}} \prod_{v\in V(F)} \left( \sum_{\substack{v'\in V(F)\\ v'\succeq v}} n_{v'} \right)^{-\alpha_v},
$$
where $\alpha_v=d_F(v)$. Ladder trees reproduce the classical stuffle multiple zeta value formula after the standard change of variables, and for convergent forests one has
$$
\zeta^t(F)=\zeta^T(s^T(F)).
$$
This framework also introduces the yew product $I$, which is commutative and not associative in general, and a second product ${}^t$, which is associative, commutative, has the empty forest as unit, preserves convergence, and makes tree zeta values multiplicative. The same paper applies the series representation to Mordell–Tornheim zeta values and to conical zeta values, characterizing a class of tree-like unimodular cones whose values are rational linear combinations of multiple zeta values [2202.00611].

A separate line of work addresses Manchon’s question of a natural tree-level map between the series-side and integral-side arborifications. The trivial map $\ell_X\circ s\circ a_Y$ commutes but collapses tree geometry to ladders; the later construction instead passes to planar rooted trees and defines a corrected map
$$
s^{PT}:H_{NBCK}^{PY}\to H_{NBCK}^{PX}
$$
by recursively correcting the defect of Clavier’s natural candidate $s^{PN}$. The main theorem proves
$$
a_{PX}\circ s^{PT}=s\circ a_{PY},
$$
and averaging over planar orders then yields a map
$$
s^T:H_{BCK}^Y\to H_{BCK}^X
$$
for ordinary rooted trees with the same commutative-square property. This settles the compatibility problem at tree level without forcing all trees to become ladders [2508.20387].

Another extension recasts the zeta constructions in tridendriform and dendriform terms using Schröder trees and binary trees. Formal series and formal integrals are first organized into tridendriform and dendriform algebras, respectively; universal properties of free tridendriform and dendriform algebras then produce zeta maps on trees. The resulting arborified zeta values are algebra morphisms for associative products on decorated Schröder trees, flatten back to classical multiple zeta values, and, on the integral side, a family of them is represented as Shintani zeta values $\zeta_{A_t}(\omega_t)$ determined by explicit combinatorial data of the tree [2508.19863].

These developments clarify three points. First, the word case is the ladder subcase, not the generic one. Second, the relation between series and integral arborifications is structurally nontrivial and requires explicit tree-level correction mechanisms. Third, arborified zeta values now sit inside a broader landscape linking rooted forests with Mordell–Tornheim, conical, and Shintani-type zeta theories, while preserving the guiding principle that tree combinatorics controls both algebraic identities and analytic realizations [2202.00611][2508.20387][2508.19863].

Source: https://www.emergentmind.com/topics/arborified-zeta-values