Epichristoffel Tree Overview

• Epichristoffel tree is a binary structure that recursively organizes epichristoffel words, generalizing classical Christoffel words through episturmian morphisms for alphabets of size k ≥ 3.
• It distinguishes a subclass of epichristoffel words that preserve a Christoffel-like factorization, contrasting with the unique factorization seen in the binary case.
• The tree incorporates a tuple structure analogous to the Stern–Brocot tree, using vector addition to propagate occurrence counts and establish existence results.

Searching arXiv for the specified paper and closely related context. The epichristoffel tree is a binary tree introduced as a direct generalization of the classical Christoffel tree from binary words to epichristoffel words over an alphabet of size $k \ge 3$. Its role is to organize epichristoffel words in a recursive structure analogous to the Christoffel/Stern–Brocot tree and to isolate a subclass of epichristoffel words that admit a factorization into two smaller epichristoffel words (Krishnamoorthy et al., 21 Jul 2025). In the underlying development, the tree serves simultaneously as a combinatorial construction, a factorization device, and a framework for existence results concerning epichristoffel words and their occurrence tuples.

1. Classical background and the generalization problem

Classical Christoffel words are finite binary words over $\{x,y\}$ that arise geometrically as encodings of discrete lattice paths approximating a line segment of rational slope and algebraically as balanced Lyndon words. A lower Christoffel path of slope $\frac{a}{b}$ is the lattice path from $(0,0)$ to $(b,a)$ that stays below the segment joining $(0,0)$ and $(b,a)$ and encloses no lattice points other than those on the path. Reading each horizontal step as $x$ and each vertical step as $y$ yields a Christoffel word. The standard characterization recalled in the paper is

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$

A central structural property of Christoffel words is that every nontrivial Christoffel word admits a unique standard factorization into two Christoffel words. This gives rise to the classical Christoffel tree, whose nodes are pairs $\{x,y\}$0, whose represented word is $\{x,y\}$1, whose children are

$\{x,y\}$2

and whose root is $\{x,y\}$3. A classical result recalled in the paper is that every Christoffel word appears exactly once in this tree, in standard factorization form (Krishnamoorthy et al., 21 Jul 2025).

The epichristoffel tree is introduced to extend this paradigm beyond the binary setting. The motivating difficulty is that many properties of Christoffel words carry over to epichristoffel words, but many do not. In particular, the binary theory has a universal factorization property, whereas in the epichristoffel setting factorization into two smaller epichristoffel words is no longer automatic. The tree is therefore not merely an analogue of a known object; it is also a method for determining which epichristoffel words retain a Christoffel-like factorization behavior.

2. Epichristoffel words and episturmian morphisms

The construction is formulated for epichristoffel words, introduced by Paquin as a generalization of Christoffel words to alphabets of size $\{x,y\}$4 (Krishnamoorthy et al., 21 Jul 2025). Their definition is expressed through episturmian morphisms.

For an alphabet $\{x,y\}$5 and letters $\{x,y\}$6, the morphisms recalled in the paper are: $\{x,y\}$7

$\{x,y\}$8

together with the exchange morphism $\{x,y\}$9, which swaps $\frac{a}{b}$0 and fixes the other letters. The episturmian morphisms form the monoid generated by these maps, and the pure episturmian morphisms form the submonoid generated by the $\frac{a}{b}$1.

A finite word $\frac{a}{b}$2 is in an epichristoffel class if it is the image of a letter under an episturmian morphism. An epichristoffel word is the unique Lyndon word in such a class. A word conjugate to an epichristoffel word is called c-epichristoffel. For an epichristoffel word $\frac{a}{b}$3 over $\frac{a}{b}$4, the paper defines the epichristoffel $\frac{a}{b}$5-tuple

$\frac{a}{b}$6

An existence criterion for such words is recalled via a tuple-reduction operator $\frac{a}{b}$7: $\frac{a}{b}$8 where $\frac{a}{b}$9 is the chosen maximal component. The cited result states that there exists an epichristoffel word with occurrence vector $(0,0)$0 if and only if iterating $(0,0)$1 eventually yields a tuple with exactly one coordinate equal to $(0,0)$2 and all others $(0,0)$3 (Krishnamoorthy et al., 21 Jul 2025). This criterion plays the role of a gcd-type test in the generalized setting.

A further structural lemma underlies the iterative construction of words from tuples: if $(0,0)$4 is c-epichristoffel, then there exists a c-epichristoffel word $(0,0)$5 with $(0,0)$6 and an episturmian morphism $(0,0)$7 such that

$(0,0)$8

if and only if

$(0,0)$9

This provides the recursive mechanism needed for the tree construction.

3. Definition and construction of the epichristoffel tree

The epichristoffel tree is introduced as a binary tree built from a chosen epichristoffel word. The starting point is an epichristoffel word $(b,a)$0 of length $(b,a)$1, with associated tuple $(b,a)$2. Using the iterative episturmian construction, one factors a c-epichristoffel representative of $(b,a)$3 as

$(b,a)$4

where both $(b,a)$5 and $(b,a)$6 are c-epichristoffel. If the tuples of $(b,a)$7 and $(b,a)$8 are $(b,a)$9 and $(0,0)$0, one then chooses the lexicographically smallest conjugate of $(0,0)$1 as the epichristoffel word at the root and sets the root node to be $(0,0)$2, where $(0,0)$3 and $(0,0)$4 are the epichristoffel words corresponding to the two factors (Krishnamoorthy et al., 21 Jul 2025).

The parent-child rule is formally identical to that of the Christoffel tree: $(0,0)$5 The key structural input is Lemma 5.1, which states that if $(0,0)$6 is a c-epichristoffel word whose factorization is obtained by the episturmian construction, then

$(0,0)$7

are also c-epichristoffel words. Consequently, if $(0,0)$8 is c-epichristoffel, both children remain inside the c-epichristoffel class; by taking lexicographically smallest representatives of conjugacy classes, the descendants are again epichristoffel.

This closure is formalized in Theorem 5.1: every word appearing in the epichristoffel tree is obtained by finitely many episturmian morphisms from a letter and is lexicographically smallest in its conjugacy class; hence every word appearing in the tree is epichristoffel (Krishnamoorthy et al., 21 Jul 2025). The tree therefore contains only epichristoffel words, not merely c-epichristoffel representatives.

Unlike the classical Christoffel tree, the epichristoffel tree is rooted at a chosen epichristoffel word and is not claimed to contain all epichristoffel words over a $(0,0)$9-letter alphabet. The construction instead yields a structured subclass whose internal behavior reflects the factorization and tuple dynamics of the selected root.

4. Factorization: preservation, obstruction, and subclasses

The factorization problem is the main structural contrast between Christoffel and epichristoffel words. In the binary case, every Christoffel word can be written as a product of two Christoffel words. In the generalized case, this fails: an epichristoffel word cannot always be written as the product of two epichristoffel words. The paper gives the explicit example

$(b,a)$0

over $(b,a)$1, which has no such factorization (Krishnamoorthy et al., 21 Jul 2025).

At the level of conjugacy classes, however, a weaker property persists. Any c-epichristoffel word $(b,a)$2 of length $(b,a)$3 can be non-uniquely written as a product of two c-epichristoffel words. This nonunique factorization is the combinatorial fact that makes the epichristoffel tree possible.

One of the principal contributions of the tree is that it separates epichristoffel words into two subclasses: those that can be factored into two epichristoffel words and those that cannot. The two corresponding preservation results are asymmetric but strong.

If the root word $(b,a)$4 cannot be factorized as a product of two epichristoffel words, then the nonfactorizability persists along the rightmost diagonal. Theorem 5.4 states that if $(b,a)$5 is an epichristoffel word that cannot be factorized as the product of two epichristoffel words, then the right diagonal $(b,a)$6 of the epichristoffel tree formed with $(b,a)$7 as root contains epichristoffel words that cannot be factorized as a product of two epichristoffel words (Krishnamoorthy et al., 21 Jul 2025). The proof mechanism described in the paper is that if the root is $(b,a)$8 with $(b,a)$9 not epichristoffel, then the right diagonal consists of

$x$0

so the second component remains non-epichristoffel.

Conversely, Theorem 5.5 states that if $x$1 is an epichristoffel word that can be factorized as the product of two epichristoffel words, then every word in the epichristoffel tree formed with $x$2 as root can also be factorized as the product of two epichristoffel words (Krishnamoorthy et al., 21 Jul 2025). Thus the tree preserves positive factorability globally once it is present at the root.

This division into factorable and nonfactorable subclasses is one of the clearest ways in which the epichristoffel tree differs from the Christoffel tree. In the classical binary setting, the factorization property is universal and unique; in the generalized setting, the tree becomes a diagnostic framework for determining where the Christoffel-like decomposition survives.

5. Tuple structure and the Stern–Brocot analogue

The paper associates to the epichristoffel tree a “Stern–Brocot tree of tuples.” This correspondence mirrors the relation between the Christoffel tree and the classical Stern–Brocot tree of fractions, but the mediant operation is replaced by vector addition.

For two $x$3-tuples

$x$4

their mediant is defined as

$x$5

Starting from the two root tuples corresponding to $x$6 and $x$7, mediants are inserted between consecutive terms level by level, exactly as in the classical Stern–Brocot construction (Krishnamoorthy et al., 21 Jul 2025). The paper emphasizes that the same diagonal behavior from the Stern–Brocot tree persists in this tuple setting.

The associated tuple tree is not merely a formal analogue. It supports concrete existence results and gives a systematic way to propagate occurrence vectors through the tree. This is reflected in Theorem 5.2, which states that if $x$8 is an epichristoffel word corresponding to the tuple $x$9, then there exist epichristoffel words for tuples of the form

$y$0

for some positive integers $y$1 (Krishnamoorthy et al., 21 Jul 2025). The stated interpretation is that this is an epichristoffel analogue of a classical diagonal property of the Stern–Brocot tree: later diagonals are sums of earlier diagonals.

The paper uses this mechanism to derive a length-existence theorem over a 3-letter alphabet. From the tree rooted at tuple $y$2, the initial left diagonals $y$3 produce lengths of the form

$y$4

and the right diagonals $y$5 produce lengths of the form

$y$6

Combined with the existence of even lengths, this yields Theorem 5.3: for the 3-letter alphabet $y$7, there exist epichristoffel words containing each letter at least once of every length $y$8, except $y$9 (Krishnamoorthy et al., 21 Jul 2025). The tree therefore functions as an existence engine as well as a classification device.

6. Worked examples and structural significance

Several worked examples in the paper illustrate how the tree is constructed and how it is used to recover lexicographically minimal representatives of conjugacy classes.

For tuple $$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$0 over $$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$1, the iterative construction gives the c-epichristoffel word

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$2

and its lexicographically smallest conjugate

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$3

which is the epichristoffel word (Krishnamoorthy et al., 21 Jul 2025). This example shows the passage from a c-epichristoffel representative to the distinguished Lyndon representative.

For tuple $$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$4, the construction yields

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$5

with corresponding tuples $$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$6 and $$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$7, and the epichristoffel word

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$8

The tree is rooted at

$$\text{A word } w \text{ is Christoffel } \Longleftrightarrow w \text{ is a balanced Lyndon word.}$$9

with children obtained by

$\{x,y\}$00

The associated tuple tree begins with

$\{x,y\}$01

$\{x,y\}$02

$\{x,y\}$03

and so on (Krishnamoorthy et al., 21 Jul 2025). This same root also illustrates the nonfactorizable subclass: the right diagonal yields tuples

$\{x,y\}$04

all lying in the nonfactorizable subclass.

For tuple $\{x,y\}$05, the c-epichristoffel word is

$\{x,y\}$06

which is already the epichristoffel word, and the corresponding tree has root

$\{x,y\}$07

This example exhibits the factorable case in which the root decomposition already consists of epichristoffel words.

A further example addresses the problem of identifying the epichristoffel word in a conjugacy class. For tuple $\{x,y\}$08, following the path

$\{x,y\}$09

in the epichristoffel tree yields the epichristoffel word

$\{x,y\}$10

The paper uses this to answer Paquin’s question about characterizing the epichristoffel word in a conjugacy class: one finds the tuple’s node in the epichristoffel tree and reads off the corresponding lexicographically minimal representative (Krishnamoorthy et al., 21 Jul 2025).

Taken together, these examples show that the epichristoffel tree is not only an abstract generalization of the Christoffel tree. It is also a concrete computational framework for constructing words from tuples, locating canonical representatives within conjugacy classes, and distinguishing between factorable and nonfactorable behavior.

7. Similarities and differences with the Christoffel tree

The epichristoffel tree retains several defining features of the classical Christoffel tree. Both are infinite binary trees. Both use nodes represented as pairs $\{x,y\}$11. Both apply the same recursive child rule,

$\{x,y\}$12

Both are linked to a Stern–Brocot-type structure via a mediant operation. Both encode a canonical decomposition process, and both are used to enumerate and characterize distinguished words in combinatorics on words (Krishnamoorthy et al., 21 Jul 2025).

The differences are equally fundamental. Christoffel trees are binary and associated with rational slopes or fractions, whereas epichristoffel trees work with $\{x,y\}$13-tuples and episturmian morphisms. Christoffel words always factor uniquely into two Christoffel words; epichristoffel words do not always factor into two epichristoffel words. The epichristoffel tree depends on a chosen epichristoffel root word, and different roots can yield different trees. It is not claimed to contain all epichristoffel words over a $\{x,y\}$14-letter alphabet; rather, it contains a structured subclass. Finally, the tuple mediant is vector addition,

$\{x,y\}$15

not fraction mediant.

These contrasts define the conceptual status of the epichristoffel tree. It transfers the intuition of Christoffel theory to the higher-alphabet setting, but it does so in a form adapted to the additional complexity of episturmian morphisms, conjugacy classes, and nonuniversal factorization. A plausible implication is that the tree should be viewed less as a complete enumeration device than as a structural framework for the part of epichristoffel theory that still behaves in a Christoffel-like manner.