Nonsimple Butterfly Trees

• Nonsimple butterfly trees are recursively defined rooted-tree structures arising from butterfly permutations and constructed via independent local choices in nonsimple butterfly matrices.
• They connect algebraic formulations with probabilistic properties such as BST height, longest increasing subsequences, and cycle counts that scale with power laws.
• Empirical studies reveal that nonsimple butterfly trees demonstrate rigid Horton–Strahler behavior and distinct differences from simple butterfly trees and uniform random permutations.

Searching arXiv for the relevant butterfly-tree literature to ground the article in current papers. arXiv search query: butterfly trees nonsimple butterfly permutations Horton-Strahler number heights butterfly trees “Nonsimple butterfly trees” denotes, in the context of butterfly permutations, the rooted-tree structure underlying the nonsimple butterfly groups $B_n^{(p)}$ and the random permutations drawn from them. In the primary formulation, these objects arise from Gaussian elimination with partial pivoting on nonsimple scalar butterfly matrices, whose permutation factors form iterated wreath products and, for prime $p$, realize $p$-Sylow subgroups of $S_{p^n}$ (Peca-Medlin et al., 2024). In later work on binary search trees and Horton–Strahler complexity, the same recursive permutation class induces “general” or “nonsimple” butterfly trees as binary tree shapes obtained from the associated permutations, with markedly different behavior from both simple butterfly trees and classical random trees (Peca-Medlin et al., 6 Jul 2025, Peca-Medlin, 14 Sep 2025).

1. Algebraic origin and basic definition

For $N=2^n$, a scalar butterfly matrix is built recursively as

$B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$

with

$$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$

and $A_1,A_2$ butterfly matrices of size $N/2$. Applying GEPP produces a permutation factor $\sigma(B)\in S_N$, called a butterfly permutation (Peca-Medlin et al., 2024).

The simple–nonsimple distinction is structural. Simple scalar butterfly matrices satisfy $p$0 recursively and can be written as

$p$1

Their permutation group is

$p$2

Nonsimple scalar butterfly matrices allow $p$3 and $p$4 to differ, and their permutation set forms a group

$p$5

Hence

$p$6

an $p$7-fold wreath product, with $p$8 normal in $p$9 (Peca-Medlin et al., 2024).

For general base $p$0, the same pattern persists: $p$1 When $p$2 is prime,

$p$3

so these groups are $p$4-Sylow subgroups of $p$5 (Peca-Medlin et al., 2024).

2. Rooted-tree interpretation

The rooted-tree viewpoint is the most direct meaning of “nonsimple butterfly trees.” The structure of $p$6 is exactly the group of automorphisms of a rooted $p$7-ary tree of depth $p$8. The vertices are words of length $p$9 over $S_{p^n}$0, and each level corresponds to one digit in base $S_{p^n}$1. An element of $S_{p^n}$2 acts by independently permuting depth-1 subtrees via elements of $S_{p^n}$3, followed by a cyclic permutation of the $S_{p^n}$4 children through $S_{p^n}$5 (Peca-Medlin et al., 2024).

In the binary case, the tree is a full binary tree of depth $S_{p^n}$6. At each node, one either keeps the left and right subtrees in place or swaps them. The essential difference from the simple model is locality. In the simple group $S_{p^n}$7, the choice at depth $S_{p^n}$8 is global: all nodes at that depth use the same transposition or identity. In the nonsimple group $S_{p^n}$9, each node chooses independently, producing a much larger group and more intricate permutations (Peca-Medlin et al., 2024).

Object	Group structure	Tree interpretation
Simple butterfly	$N=2^n$0	one global choice per level
Nonsimple butterfly	iterated wreath product $N=2^n$1	local choices at each node

This tree interpretation is also the bridge to later binary-search-tree models. Given a butterfly permutation $N=2^n$2, one may form the BST $N=2^n$3 by inserting the values of $N=2^n$4 in order. For butterfly permutations, the resulting binary tree shape is called a butterfly tree; simple trees arise from iterated gluing of identical copies, whereas nonsimple butterfly trees arise by gluing different butterfly trees at successive levels (Peca-Medlin, 14 Sep 2025).

3. Random generation, uniformity, and Sylow subgroups

The random nonsimple model is tied to GEPP. If $N=2^n$5 is a nonsimple scalar butterfly matrix built with independent uniform angles $N=2^n$6 at each level, then

$N=2^n$7

Thus GEPP on such matrices produces a Haar-uniform element of the nonsimple butterfly group $N=2^n$8 (Peca-Medlin et al., 2024).

This uniformity rests on an explicit block GEPP factorization,

$N=2^n$9

combined with the recursive wreath-product structure and the Diaconis–Shahshahani subgroup algorithm (Peca-Medlin et al., 2024).

For prime $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$0, the $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$1-ary formulation is expressed directly in group-theoretic terms: $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$2 is a uniform random element of a $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$3-Sylow subgroup of $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$4. Because all $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$5-Sylow subgroups of $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$6 are conjugate and the number of cycles is conjugation-invariant, distributional results for $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$7 transfer immediately to any uniform $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$8-Sylow subgroup (Peca-Medlin et al., 2024).

A recurrent misconception is to identify these random objects with uniform permutations in $B=(R_\theta\otimes I_{N/2})(A_1\oplus A_2),$9. They are instead highly constrained hierarchical permutations. This distinction is decisive for longest increasing subsequences, cycle counts, tree height, and Horton–Strahler behavior.

4. Longest increasing subsequences, cycles, and height

For nonsimple permutations $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$0, the longest increasing subsequence $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$1 is encoded by a recursive process $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$2 with $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$3 and

$$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$4

where the $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$5 are iid copies of $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$6. In the binary case,

$$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$7

This is the tree recursion in probabilistic form: if the root permutation is identity, LIS contributions from both subtrees can concatenate; if the root swaps blocks, monotonicity forces the LIS into one side (Peca-Medlin et al., 2024).

The expected LIS obeys a power law. If $$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$8, then there exist constants

$$R_\theta=\begin{bmatrix}\cos\theta & \sin\theta\ -\sin\theta & \cos\theta\end{bmatrix},$$9

such that

$A_1,A_2$0

and $A_1,A_2$1. In the binary case,

$A_1,A_2$2

and numerical estimation gives $A_1,A_2$3 (Peca-Medlin et al., 2024). This contrasts with uniform permutations, for which $A_1,A_2$4.

The cycle count is even more explicitly tree-recursive. In the binary nonsimple case, if $A_1,A_2$5, then

$A_1,A_2$6

where $A_1,A_2$7 are iid and $A_1,A_2$8. Hence

$A_1,A_2$9

With

$N/2$0

one has

$N/2$1

where $N/2$2 has support $N/2$3 and integer moments determined by

$N/2$4

For prime $N/2$5, the normalization becomes

$N/2$6

with $N/2$7 supported on $N/2$8 and uniquely determined by its recursively defined moments (Peca-Medlin et al., 2024).

The BST height induced by nonsimple butterfly permutations also grows polynomially. If $N/2$9 and $\sigma(B)\in S_N$0 is the associated BST, then

$\sigma(B)\in S_N$1

with

$\sigma(B)\in S_N$2

The lower bound matches the simple butterfly mean height, while the upper bound reflects the max-type recursion in the wreath-product model (Peca-Medlin et al., 6 Jul 2025).

5. Horton–Strahler number and rigidity in the nonsimple model

The Horton–Strahler number $\sigma(B)\in S_N$3 provides a distinct tree-complexity statistic. For a rooted binary tree, it is defined recursively by

$\sigma(B)\in S_N$4

For butterfly trees, $\sigma(B)\in S_N$5 measures how deeply perfect binary branching can be embedded in the tree shape (Peca-Medlin, 14 Sep 2025).

In the nonsimple case, exact analysis remains open, but one rigorous global bound is known: for any butterfly tree $\sigma(B)\in S_N$6 with $\sigma(B)\in S_N$7 nodes,

$\sigma(B)\in S_N$8

and this bound is sharp (Peca-Medlin, 14 Sep 2025). This support bound is substantially smaller than the maximum possible Horton–Strahler number for arbitrary binary trees on $\sigma(B)\in S_N$9 nodes.

The obstruction to a simple theory is structural. In the simple model, the gluing of identical subtrees collapses the $p$00 evolution to an 8-state Markov chain. In the general nonsimple model, one must track how maximal $p$01 paths run along both top edges, including escape information into internal subtrees; the natural state becomes an evolving combinatorial object whose state space grows with tree size, and no finite-state Markov representation is known (Peca-Medlin, 14 Sep 2025).

Empirical sampling nevertheless shows striking concentration. For $p$02, 10,000 uniform nonsimple butterfly trees yielded the following observed $p$03 distribution (Peca-Medlin, 14 Sep 2025):

HS	count
3	494
4	9,040
5	466

For the same experiment, the empirical mean was $p$04, and the empirical variance was $p$05 with standard deviation $p$06 (Peca-Medlin, 14 Sep 2025). Since $p$07, this suggests concentration tightly below the structural upper bound. A plausible implication is that nonsimple butterfly trees are more rigid in $p$08 than simple butterfly trees, even though they are larger and combinatorially more varied.

6. Comparisons, alternate usages, and open problems

The contrast with simple butterflies is sharp. Simple butterfly permutations form an abelian group $p$09, corresponding to “one global choice per level,” and in the binary BST setting they produce an exact height law

$p$10

with

$p$11

(Peca-Medlin et al., 6 Jul 2025). Nonsimple butterflies replace this exact tensor-product structure by an iterated wreath product, yielding only bounds for height and for LIS exponents, but a full scaling limit for cycle counts (Peca-Medlin et al., 2024, Peca-Medlin et al., 6 Jul 2025).

The contrast with uniform permutations in $p$12 is equally strong. Uniform random permutations have $p$13, Tracy–Widom LIS fluctuations, and cycle counts of order $p$14, whereas nonsimple butterfly permutations have LIS and cycle counts that scale as powers of $p$15 with exponents strictly larger than $p$16 for LIS and $p$17 for cycles (Peca-Medlin et al., 2024). This suggests that the rooted-tree hierarchy imposes long-range order rather than washing it out.

The terminology is not universal across arXiv. In “Building the Butterfly Fractal: The Eightfold Way,” a “butterfly tree” is an octonary Diophantine tree for the Hofstadter butterfly, generated by eight unimodular $p$18 integer matrices acting on $p$19; that object belongs to spectral topology rather than permutation groups or BSTs (Satija, 2024). In “Superconcentration on a Pair of Butterflies,” the relevant tree-like object is the binary hierarchy of sub-butterflies in a layered routing network, where a $p$20-dimensional butterfly recursively splits into $p$21, $p$22, then $p$23, and so on (Bradley, 2014). These usages are mathematically distinct from nonsimple butterfly trees arising from $p$24.

Several central problems remain open. For general nonsimple butterfly trees, the order of growth and limit law of the Horton–Strahler number are unproved; the variance may be bounded, logarithmic, or intermediate, and the empirical concentration near $p$25 lacks a rigorous mechanism (Peca-Medlin, 14 Sep 2025). For BST height, identifying a single exponent $p$26 such that $p$27 remains open (Peca-Medlin et al., 6 Jul 2025). For the permutation model more broadly, the LIS is controlled only by power-law bounds rather than an exact exponent or limit law (Peca-Medlin et al., 2024).

Taken together, these results place nonsimple butterfly trees in an intermediate regime between maximally structured simple butterflies and fully random symmetric-group behavior. Their governing principle is recursive local randomness on a rooted tree: sufficiently rigid to force anomalously large LIS, cycle counts, and BST heights, yet sufficiently rich to defeat the finite-state methods that solve the simple case.