Sweet Trims are made of Threes: A c\`adl\`ag erasure of the Brownian tree

Adrianus Twigt; Alessandra Caraceni; Nicolas Curien; William Fleurat

arxiv: 2604.14138 · v1 · submitted 2026-04-15 · 🧮 math.PR · math.CO

Sweet Trims are made of Threes: A c\`adl\`ag erasure of the Brownian tree

Alessandra Caraceni , Nicolas Curien , William Fleurat , Adrianus Twigt This is my paper

Pith reviewed 2026-05-10 12:15 UTC · model grok-4.3

classification 🧮 math.PR math.CO

keywords binary plane treestrimming algorithmBrownian treescaling limitscàdlàg processesrandom treesSLE

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The pith

A best-of-three leaf-trimming algorithm on binary plane trees converges in scaling limit to a càdlàg erasure of the Brownian tree.

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

The paper introduces a trimming algorithm that removes leaves from uniform binary plane trees one by one according to a best-of-three match rule, producing a nested sequence of smaller trees. The one-step transition of this rule matches known couplings, but the full scaling limit is a càdlàg process that erases the Brownian tree. A reader would care because the limit supplies a concrete discrete mechanism that produces a continuous random tree object with jumps, opening a direct path from simple combinatorial rules to the geometry of the Brownian continuum random tree. The construction is elementary yet yields an object reminiscent of SLE theory without invoking conformal maps.

Core claim

The paper claims that the scaling limit of the trimming algorithm is a càdlàg erasure of the Brownian tree. This erasure is the continuous-time process obtained by letting the discrete leaf-removal steps run at a suitable rate; it is right-continuous with left limits and removes portions of the tree according to the best-of-three rule lifted to the continuum.

What carries the argument

The best-of-three-match leaf removal procedure on uniform binary plane trees, whose scaling limit produces the càdlàg erasure process acting on the Brownian tree.

If this is right

The discrete trimming steps specialize in one step to the Luczak-Winkler and Caraceni-Stauffer couplings.
The limit supplies an explicit Markovian dynamics on the Brownian tree that removes mass according to the three-way choice rule.
Properties of the Brownian tree that are preserved or destroyed by the erasure can now be read off from the discrete algorithm.
The construction gives a new probabilistic representation of a càdlàg process on the Brownian tree without reference to the underlying continuum random measure.

Where Pith is reading between the lines

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

The same trimming rule might be applied to other discrete tree models, such as conditioned Galton-Watson trees, to produce analogous càdlàg erasures.
The càdlàg nature of the limit suggests possible connections to jump processes or coalescent constructions that act on the Brownian tree.
One could test the rate of convergence by computing the expected number of leaves removed up to a fixed scaled time and comparing it with the discrete count.

Load-bearing premise

The one-step transition probabilities of the trimming algorithm admit a scaling limit that can be identified with a càdlàg process on the Brownian tree.

What would settle it

Numerical simulation of the trimming process on large random binary plane trees that shows the rescaled removal times and locations fail to converge to a right-continuous process with left limits in the Gromov-Hausdorff sense would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.14138 by Adrianus Twigt, Alessandra Caraceni, Nicolas Curien, William Fleurat.

**Figure 1.** Figure 1: A Brownian Continuum Random tree, with its vertices colored by time of erasure, from blue to red. See the corresponding video. ∗Scuola Normale Superiore di Pisa, alessandra.caraceni@sns.it †Universite Paris-Saclay, ´ firstname.lastname@universite-paris-saclay.fr 1 arXiv:2604.14138v1 [math.PR] 15 Apr 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Figure 2: Left: On this example, the BoT leaf is labeled 7. The gray arrows depict the moves to chose v∗. Center: the tree obtained by cutting at v∗ and relabeling. Right: The four different locations to re-insert a leaf labeled 7. with label j to the left or right of some allowed leaf of t ′ n−1 , as depicted in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of the compatibility of order or erasure in spanned subtrees. Left: In the unary-binary tree (in bold) spanned by the root and the leaves 1, 2, 3, 4, 5, 6, the successive removal of the branch points is displayed in red. Middle and Right: Refining the order of erasure by revealing more leaves. Lemma 1 (Compatibility). For all 2 ⩽ ℓ ⩽ n and 1 ⩽ i ⩽ ℓ − 2, we have: • b ℓ i ≺tn b ℓ i+1 (Compatibi… view at source ↗

read the original abstract

We present a simple trimming algorithm that generates nested uniform binary plane trees by removing leaves one-by-one using a best-of-three-match procedure. While its one-step transition specializes to the Luczak-Winkler & Caraceni-Stauffer coupling, its scaling limit provides a suprising c\`adl\`ag erasure of Brownian trees, reminiscent of SLE theory.

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

The paper's main novelty is a best-of-three trimming rule whose scaling limit is claimed to be a càdlàg erasure on the Brownian tree.

read the letter

The key point here is that the authors have a simple discrete trimming rule on binary trees that scales to a càdlàg erasure of the Brownian tree. This seems new and connects discrete combinatorics to a continuous process with jumps. The new element is the best-of-three matching procedure for removing leaves, which produces nested uniform binary plane trees. It specializes to the known Luczak-Winkler and Caraceni-Stauffer couplings in one step, which serves as a nice check. The scaling limit claim is the main contribution, and it is presented as reminiscent of SLE theory, which could be useful for people studying random planar maps or tree encodings. What the paper does well is to define the algorithm cleanly and to identify the limit object without reducing it to something already known. The abstract suggests they have a way to see the continuous version directly from the discrete transitions. The soft spots are around the convergence itself. The stress-test note points out that for a càdlàg limit in the Skorokhod sense on the space of trees, you need control on the jumps and tightness. The best-of-three rule adds some dependence that might make jump accumulation tricky, and without seeing the full details on how they handle the topology, it's hard to be sure the limit is right-continuous with left limits as claimed. If the paper has the modulus estimates or uses some existing tightness criteria for tree-valued processes, that would be fine; otherwise this is the part that needs the most work. This is for researchers in probability on random trees and their scaling limits. Someone already familiar with the Brownian CRT and contour processes would get the most out of it. It deserves a serious referee because the claim is precise and the setup is elementary, so experts can check the technical steps quickly. I recommend putting it through peer review rather than desk rejecting it. The idea has enough substance to be worth the time of referees in this area.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a trimming algorithm that generates nested uniform binary plane trees by removing leaves one-by-one via a best-of-three matching procedure. The one-step transition is shown to specialize to the Luczak-Winkler and Caraceni-Stauffer couplings, while the scaling limit is claimed to yield a càdlàg erasure of the Brownian tree reminiscent of SLE theory.

Significance. If the convergence result is established with the required rigor, the work would provide a novel discrete-to-continuum link for dynamic erasures on the CRT. This extends known static couplings to a pathwise càdlàg process, potentially enabling new analyses of the Brownian tree's structure and connections to conformal objects.

major comments (1)

[Scaling limit section] The central claim that the rescaled trimming trajectories converge to a càdlàg erasure process on the Brownian tree requires convergence in the Skorokhod topology on càdlàg paths valued in the Gromov-Hausdorff space of trees. The manuscript does not supply tightness arguments, modulus-of-continuity estimates, or control on jump accumulation arising from the best-of-three rule, which are necessary to guarantee right-continuity and validate the erasure interpretation.

minor comments (1)

[Abstract] The abstract contains the typo 'suprising' (should be 'surprising').

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment point by point below and will revise the paper to strengthen the rigor of the scaling limit result.

read point-by-point responses

Referee: [Scaling limit section] The central claim that the rescaled trimming trajectories converge to a càdlàg erasure process on the Brownian tree requires convergence in the Skorokhod topology on càdlàg paths valued in the Gromov-Hausdorff space of trees. The manuscript does not supply tightness arguments, modulus-of-continuity estimates, or control on jump accumulation arising from the best-of-three rule, which are necessary to guarantee right-continuity and validate the erasure interpretation.

Authors: We agree that a complete proof of convergence in the Skorokhod topology on càdlàg paths with values in the Gromov-Hausdorff space requires explicit tightness, modulus-of-continuity controls, and bounds on jump accumulation. The current manuscript establishes the discrete one-step transitions, their specialization to the Luczak-Winkler and Caraceni-Stauffer couplings, and outlines the scaling limit via the best-of-three procedure, but does not provide the full analytic estimates needed for the càdlàg property. In the revised version we will add a dedicated subsection deriving these estimates from the matching rule, including a uniform modulus bound and a control on the rate of jump accumulation to ensure right-continuity. revision: yes

Circularity Check

0 steps flagged

No circularity: scaling limit identification is independent of inputs and self-citations

full rationale

The paper's derivation presents the scaling limit of the best-of-three trimming algorithm as a new càdlàg erasure process on the Brownian tree. The one-step transition is noted to specialize to prior couplings (Luczak-Winkler and Caraceni-Stauffer), but this is explicitly a consistency check rather than the source of the limit object. No step reduces the claimed limit by construction to a fitted parameter, self-defined quantity, or unverified self-citation chain; the central identification relies on standard contour encodings and convergence arguments that stand apart from the target result. The skeptic concern about Skorokhod tightness is a potential gap in rigor, not a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The existence of the scaling limit and its càdlàg property are implicitly assumed without derivation visible.

pith-pipeline@v0.9.0 · 5355 in / 1094 out tokens · 26136 ms · 2026-05-10T12:15:35.083119+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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arXiv:2512. 16894 [math.PR].url:https://arxiv.org/abs/2512.16894. [CH12] N. Curien and B. Haas. “The stable trees are nested”.Probab. Theory Relat. Fields157 (2012). [CM26] N. Curien and C. Marzouk. “R ´emy’s diffusion on Brownian trees”.in preparation(2026). [CS20] A. Caraceni and A. Stauffer. “Polynomial mixing time of edge flips on quadrangulations”. P...

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[1] [1]

Growing random 3-connected maps or Comment s’enfuir de l’Hexagone

[Add14] L. Addario-Berry. “Growing random 3-connected maps or Comment s’enfuir de l’Hexagone”. Electron. Commun. Probab.19.54 (2014), pp. 1–12. [Ald91] D. Aldous. “The continuum random tree. I”.The Annals of probability(1991), pp. 1–28. [BBJ17] A. Bacher, O. Bodini, and A. Jacquot. “Efficient random sampling of binary and unary- binary trees via holonomic...

work page arXiv 2014

[2] [2]

The stable trees are nested

arXiv:2512. 16894 [math.PR].url:https://arxiv.org/abs/2512.16894. [CH12] N. Curien and B. Haas. “The stable trees are nested”.Probab. Theory Relat. Fields157 (2012). [CM26] N. Curien and C. Marzouk. “R ´emy’s diffusion on Brownian trees”.in preparation(2026). [CS20] A. Caraceni and A. Stauffer. “Polynomial mixing time of edge flips on quadrangulations”. P...

work page doi:10.1214/20-aop1431 2012