Rooted Tandem Repeat Trees in Genome Duplication

• Rooted Tandem Repeat Trees are defined as rooted combinatorial structures that encode the history of local tandem-duplication events, capturing both genealogy and final gene order.
• They provide a bijective mapping to counter arrays, offering an effective upper bound for the enumeration of Manacher arrays in palindromic sequence analysis.
• These trees also serve as a framework for modeling and counting distinct tandem duplication histories, aiding in the analysis of genome rearrangement and duplication events.

Rooted tandem repeat trees are rooted combinatorial structures that encode histories of local tandem-duplication events. In "Combinatorics of Palindromes" (Itzhaki, 18 Jul 2025), the term corresponds to rooted duplication trees: rooted binary trees whose root is a single ancestral gene, whose internal nodes correspond to duplication events, and whose leaves are the extant genes in left-to-right order. In "The Combinatorics of Tandem Duplication" (Penso-Dolfin et al., 2014), closely related rooted structures arise as bi-coloured 2D-trees and the rooted major graphs derived from them, where nodes represent breakpoints rather than genes. Across these settings, rooted tandem repeat trees serve as explicit encodings of duplication history and as counting devices for the combinatorics of palindromes, tandem duplication words, and breakpoint orders.

1. Gene-based rooted duplication trees

A tandem-duplication event acts on an ordered sequence of genes

$A=\{g_1,\dots,g_m\}$

by choosing a contiguous block

$$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$

and replacing the block $A[i..i+\ell-1]$ by a tandem duplication of itself using fresh descendants:

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$

where, for every $g_j\in B$, $lc(g_j)$ and $rc(g_j)$ are distinct new genes representing the left-copy and right-copy descendants of $g_j$. A block of size $\ell$ is called an $\ell$-duplication (Itzhaki, 18 Jul 2025).

Starting from a single ancestral gene,

$$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$0

successive tandem-duplication events are stored in a rooted binary tree $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$1. The root represents the ancestral gene $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$2; each internal node corresponds to one duplication event, with its left and right subtrees containing all $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$3 and $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$4 descendants created by that event; and the leaves are the genes present in the final array. The leaves are typically labeled $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$5 in left-to-right order, and this ordered list is the leaves array of $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$6. In this terminology, a rooted duplication tree with $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$7 leaves is a rooted tandem repeat tree with $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$8 genes, and $$B=\{i,i+1,\dots,i+\ell-1\}, \qquad (1\le i \le i+\ell-1 \le m),$$9 denotes the number of distinct non-isomorphic rooted duplication trees with $A[i..i+\ell-1]$0 leaves.

This model isolates the duplication history at the level of extant gene copies. The root records the unique ancestor, internal nodes record the branching imposed by each duplication event, and the leaves preserve the final linear order. The representation is therefore both genealogical and order-sensitive: it is not merely a tree of descent, but a tree whose leaves are embedded in the final tandem arrangement.

2. Enumerative role in the combinatorics of palindromes

In the palindrome setting, rooted tandem repeat trees provide an upper-bounding class for Manacher arrays. If $A[i..i+\ell-1]$1 denotes the number of distinct Manacher arrays corresponding to strings of length $A[i..i+\ell-1]$2, and $A[i..i+\ell-1]$3 denotes the number of rooted tandem duplication trees with $A[i..i+\ell-1]$4 leaves, then Theorem 3.1 states

$A[i..i+\ell-1]$5

The proof passes through counter arrays $A[i..i+\ell-1]$6, defined by the constraints

$A[i..i+\ell-1]$7

and through a compact representation of a Manacher array derived from the centers of maximal palindromic suffixes of prefixes. The number $A[i..i+\ell-1]$8 of counter arrays of length $A[i..i+\ell-1]$9 satisfies

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$0

and the compact representations of Manacher arrays form a subclass of counter arrays, yielding

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$1

(Itzhaki, 18 Jul 2025).

The cited recurrence for the number of rooted duplication trees is

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$2

and, using Stirling’s approximation, the paper notes

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$3

Combined with the lower bound

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$4

this gives

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$5

The significance is explicitly combinatorial: rooted tandem repeat trees dominate the number of Manacher arrays, so they provide a structured envelope for the diversity of palindromic profiles.

3. Event decomposition and counter-array encoding

The structural reason that rooted tandem repeat trees can be counted through arrays is a unique event decomposition. For a duplication tree $$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$6, a node is called low if and only if all of its children are leaves. Using the left-to-right leaf order, low nodes generated by the same duplication event can be grouped into blocks, yielding a unique ordered list

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$7

of duplication events such that applying the events in that order recreates the tree’s leaves. The last event is the block whose low nodes are highest in the leaf order and whose leaves form a contiguous segment; removing it produces a smaller duplication tree and an inductive decomposition (Itzhaki, 18 Jul 2025).

This decomposition gives a canonical bijection with counter arrays. If

$$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$8

then $$\{lc(g_i),\dots,lc(g_{i+\ell-1}),rc(g_i),\dots,rc(g_{i+\ell-1})\},$$9 is encoded as the strictly decreasing sequence

$g_j\in B$0

Concatenating these sequences over all events produces an array $g_j\in B$1. Inside an event, the values decrease by exactly $g_j\in B$2; between events, the ordering of events ensures $g_j\in B$3; and each entry satisfies $g_j\in B$4. Conversely, scanning a counter array from left to right and starting a new event whenever the next entry fails to be exactly $g_j\in B$5 less than the previous recovers the event blocks, the leaves array, and the duplication tree. In this way, rooted tandem repeat trees are not only counted by counter arrays; they admit a canonical linear encoding of their event history.

The same section shows how compact Manacher representations fit into this encoding. If $g_j\in B$6 is the maximal palindromic suffix of $g_j\in B$7, $g_j\in B$8 its center, and

$g_j\in B$9

then the prefix sums

$lc(g_j)$0

satisfy

$lc(g_j)$1

Defining

$lc(g_j)$2

gives

$lc(g_j)$3

so $lc(g_j)$4 is exactly a counter array. Rooted tandem repeat trees therefore enter the palindrome problem through a structural equivalence class of arrays rather than through any direct palindromic operation.

4. Breakpoint trees, 2D-trees, and major graphs

A second formalization of rooted tandem repeat trees appears in the combinatorics of tandem duplication itself. There, tandem duplication is modeled on a reference interval $lc(g_j)$5 under the assumption of unique breakpoint use: each reference position can be a breakpoint in at most one tandem duplication. The algebraic representation is a word automaton: with $lc(g_j)$6, the $lc(g_j)$7-th tandem duplication chooses indices $lc(g_j)$8 with $lc(g_j)$9 and produces

$rc(g_j)$0

The resulting word evolution records somatic connections, but multiple breakpoint layouts can yield the same word, so the paper introduces rooted tree-like breakpoint structures to encode the full ancestral relations (Penso-Dolfin et al., 2014).

The primary object is the bi-coloured 2D-tree. The roots are $rc(g_j)$1 and $rc(g_j)$2. Each tandem duplication event $rc(g_j)$3 introduces two breakpoint nodes $rc(g_j)$4 and $rc(g_j)$5, and every non-root node has two parents: one type $rc(g_j)$6 parent and one type $rc(g_j)$7 parent. For each parental pair, the edge from the parent with larger tandem-duplication number is the major edge, and the edge from the parent with smaller tandem-duplication number is the minor edge. If $rc(g_j)$8 and $rc(g_j)$9 lie on the same segment, a fence edge is added from $g_j$0 to $g_j$1. Reversing the direction of all type $g_j$2 edges and orienting fences from $g_j$3 to $g_j$4 yields a Hasse diagram for a poset on breakpoints. Restricting further to major edges and fences gives the major graph, in which every non-root node has one parent; there are therefore two rooted trees, one rooted at $g_j$5 and one at $g_j$6. Along any major chain $g_j$7, the reference positions have the single fixed linear extension

$g_j$8

In this breakpoint-based formalism, rooted tandem repeat trees are thus pairs of rooted trees plus fences, extracted from a richer two-parent ancestry graph.

5. Counting evolutions and subtree prune and graft

The major graph is the central counting object for a fixed tandem-duplication word evolution. If a node $g_j$9 has child branches with descendant counts $\ell$0, and $\ell$1 is the number of linear extensions for branch $\ell$2 alone, then without a fence the number of interleavings is

$\ell$3

If two child roots are connected by a fence and have descendant sizes $\ell$4, then the corresponding factor is

$\ell$5

Traversing the major graph and multiplying these local factors yields the number of distinct tandem-duplication evolutions consistent with the word evolution. For the example

$\ell$6

the total is

$\ell$7

(Penso-Dolfin et al., 2014).

The same paper analyzes how these rooted trees transform when a new first tandem duplication is inserted. Induced evolutions are described by 1-nodesets, equivalently $\ell$8-subtrees, which are ancestor-closed subsets containing the new first duplication and satisfying a fence condition. Given a 1-nodeset, the induced major graph is obtained by local changes of parental edges; the paper explicitly interprets these as subtree prune and graft operations in the phylogenetic sense. Summing over all induced evolutions yields the global formula

$\ell$9

with dominant growth

$\ell$0

In this sense, rooted tandem repeat trees form a super-exponentially growing configuration space of breakpoint histories.

6. Scope, limitations, and open directions

Several limitations are explicit in the palindrome application. The inequality

$\ell$1

does not assert that each rooted tandem repeat tree corresponds to a distinct Manacher array, nor that every counter array is a valid compact Manacher representation. The paper states that Manacher arrays form a strict subclass of counter arrays, and that different strings can share the same Manacher array. It also states that rooted tandem repeat trees do not appear in the graph-theoretic reconstruction framework or in the minimal-alphabet reconstruction results; there, reconstruction is reduced to proper vertex colorings of a restriction graph, whereas the duplication-tree formalism is used exclusively in the counting upper bound. Open directions include the exact number of valid Manacher arrays of length $\ell$2, a direct combinatorial characterization of valid Manacher arrays, efficient uniform sampling, and extensions to approximate palindromes or palindromes with wildcards (Itzhaki, 18 Jul 2025).

In the tandem-duplication setting, the limitations concern identifiability. A word of somatic connections does not specify exact breakpoint coordinates; a single word $\ell$3 can correspond to many 2D-trees and major trees; and a single final TD-graph is consistent with many distinct TD histories. The poset and rooted-tree machinery therefore counts ambiguity rather than removing it. The paper further notes that counting CNVs or TD-graphs themselves is much harder than counting evolutions, that deciding whether a given CNV is realizable by tandem duplication under unique breakpoint use is open, and that mixed rearrangement processes may require more general rearrangement trees (Penso-Dolfin et al., 2014).

Taken together, these two lines of work place rooted tandem repeat trees at the intersection of genome-evolution combinatorics and palindromic reconstruction. In one role they are rooted binary trees on extant genes, used to bound the space of Manacher arrays; in the other they are rooted breakpoint trees plus fences, used to enumerate tandem-duplication histories. The shared theme is that tandem repetition induces a strongly constrained but rapidly growing tree-structured state space.