About Fibonacci trees. II -- generalized Fibonacci trees

Maurice Margenstern

arxiv: 1907.04677 · v1 · pith:LIEE5XQInew · submitted 2019-07-10 · 💻 cs.DM · math.CO

About Fibonacci trees. II -- generalized Fibonacci trees

Maurice Margenstern This is my paper

Pith reviewed 2026-05-24 23:26 UTC · model grok-4.3

classification 💻 cs.DM math.CO

keywords Fibonacci treeshyperbolic plane tilingsblack node rootinggeneralized Fibonacci treestilings {p,4}tilings {p+2,3}

0 comments

The pith

Fibonacci trees rooted at a black node exhibit different properties from those rooted at a white node, but new properties of similar complexity arise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines if the tree properties tied to hyperbolic tilings {p,4} and {p+2,3} persist when the tree is rooted at a black node using the same generation rules as before. The answer is that they do not hold in the same way, but fresh properties appear that match the simplicity of the white-node case. A reader would care as this extends the study of these generalized Fibonacci trees without added complexity, building directly on the prior analysis of white-rooted versions.

Core claim

The properties of the trees associated to the tilings {p,4} and {p+2,3} of the hyperbolic plane are not the same if we consider a finitely generated tree by the same rules but rooted at a black node; however, new properties arise, no more complex than in the case of a tree rooted at a white node, and worth of interest.

What carries the argument

Generalized Fibonacci trees generated from the rules of hyperbolic tilings {p,4} and {p+2,3}, now rooted at a black node rather than white.

If this is right

The original properties from white-rooted trees do not apply directly.
New properties emerge that are comparably straightforward.
These properties merit further examination in the context of the tilings.
The extension maintains the same level of complexity as the first paper.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Color choice in node rooting affects specific tree properties but not overall structural simplicity.
Such trees might model different aspects of hyperbolic geometry depending on root selection.
Further generalizations could explore other node types or mixed rooting.

Load-bearing premise

The generation rules and the distinction between black and white nodes remain exactly as defined in the prior paper with no changes.

What would settle it

A detailed computation for a small p showing that the new properties for black-rooted trees are substantially more complex than those for white-rooted ones would disprove the claim.

read the original abstract

In this second paper, we look at the following question: are the properties of the trees associated to the tilings $\{p,4\}$ and $\{p$+$2,3\}$ of the hyperbolic plane still true if we consider a finitely generated tree by the same rules but rooted at a black node? The direct answer is no, but new properties arise, no more complex than in the case of a tree rooted at a white node, and worth of interest. The present paper is an extension of the previous paper: arXiv:1904.12135.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a straightforward extension of the author's prior work on Fibonacci trees by re-rooting at black nodes, with new but comparable properties.

read the letter

This paper is a direct sequel to the author's previous one on Fibonacci trees. It asks whether the properties hold when the tree is rooted at a black node rather than a white one, using the same rules for the tilings {p,4} and {p+2,3}. The answer is that they do not hold the same, but new properties appear. These are described as no more complex than the original ones. What the paper does well is maintain the connection to the prior work without introducing new assumptions or changing the underlying tiling construction. The extension seems clean. Where it is softer is in the lack of visible derivations in the abstract. The claim about complexity is stated but not illustrated here, so the full text would need to show explicit constructions or comparisons to make that convincing. Since the stress test found no contradiction, it probably does. The paper is aimed at researchers in discrete mathematics who study combinatorial aspects of hyperbolic geometry and tree generations. Someone following the series on Fibonacci trees would find it useful as a continuation. I think it should go to peer review. The work is grounded in the previous paper and the central claim is a straightforward observation that can be checked against the rules.

Referee Report

0 major / 2 minor

Summary. The manuscript extends the analysis of generalized Fibonacci trees associated to the hyperbolic tilings {p,4} and {p+2,3} from the prior work arXiv:1904.12135. It asks whether the previously established properties continue to hold when the finitely generated tree is rooted at a black node (rather than a white node) under the same generation rules, and concludes that the properties differ but that new properties of comparable complexity arise and merit study.

Significance. If the new black-rooted properties are derived explicitly and shown to match the complexity of the white-rooted case, the work supplies a symmetric completion to the earlier results on Fibonacci trees in hyperbolic tilings. This strengthens the combinatorial description of these objects without increasing the apparent descriptive complexity, which may be useful for enumeration or structural questions in discrete mathematics.

minor comments (2)

The abstract states that new properties 'arise' and are 'worth of interest,' but the manuscript should include at least one concrete example (e.g., a recurrence or enumeration formula) for the black-rooted case to make the claim verifiable.
The relationship to arXiv:1904.12135 is described only at a high level; a brief recap of the white-node generation rules and the black/white node distinction (perhaps in a short preliminary section) would improve readability for readers who have not consulted the prior paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We are pleased that the extension to black-rooted generalized Fibonacci trees is viewed as supplying a symmetric completion to the white-rooted case in arXiv:1904.12135 without increasing descriptive complexity.

Circularity Check

0 steps flagged

Minor self-citation to prior rules; new properties claimed independently

full rationale

The paper is explicitly an extension of arXiv:1904.12135, reusing the same generation rules and black/white node distinction without alteration. The central claim—that properties differ when rooted at a black node yet new properties remain comparably simple—is presented as a direct application yielding independent observations. No equation or derivation reduces by construction to a fitted input, self-defined quantity, or load-bearing uniqueness theorem from the cited prior work. The self-citation serves only to carry over the fixed rules, which are externally stated and not redefined here, leaving the new analysis self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are described. The work relies on definitions and rules from the referenced prior paper.

axioms (1)

domain assumption The tilings {p,4} and {p+2,3} and their associated tree-generation rules are identical to those in arXiv:1904.12135.
The paper assumes the same tilings and rules from the previous work when changing only the root color.

pith-pipeline@v0.9.0 · 5606 in / 1073 out tokens · 25577 ms · 2026-05-24T23:26:20.983846+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; phi_fixed_point echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

metallic sequences satisfy mn+2=(p-2)mn+1-mn with m0=1, m-1=0; generalization of Fibonacci sequence connected with the golden number
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking (D=3) refines

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refines
Relation between the paper passage and the cited Recognition theorem.

white/black metallic trees with rules B→BW^{p-4}, W→BW^{p-3}; preferred son property fails for black-rooted trees but successor property holds

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.