Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

Sicheng Lu; Yi Song

arxiv: 2606.21074 · v1 · pith:NT3GIOVTnew · submitted 2026-06-19 · 🧮 math.GT

Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

Sicheng Lu , Yi Song This is my paper

Pith reviewed 2026-06-26 13:08 UTC · model grok-4.3

classification 🧮 math.GT

keywords bi-colored plane treesweighted enumerationstrong Hurwitz numbersHCMU spheresmoduli spacesGromov-Hausdorff topologyalgorithmic counting

0 comments

The pith

An algorithmic count of weighted bi-colored plane trees determines strong Hurwitz numbers and connected components of HCMU sphere moduli.

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

The paper develops a unified algorithmic method to count weighted bi-colored plane trees when the numbers of black and white vertices and the total edge weights at each vertex are fixed in advance. This counting result is applied to compute the strong Hurwitz number in a special case with three branched points on Riemann spheres. The same count also gives the number of connected components in the moduli space of a special class of HCMU spheres that have one conical singularity, measured in the Gromov-Hausdorff topology.

Core claim

The enumeration of weighted bi-colored plane trees with prescribed black and white vertex counts and prescribed total edge weights per vertex admits a unified algorithmic solution. This solution directly yields the strong Hurwitz numbers for three-point branch data and the count of connected components in the moduli space of HCMU spheres with a single conical singularity.

What carries the argument

The unified algorithmic counting method for weighted bi-colored plane trees with prescribed vertex numbers and edge weights.

If this is right

Strong Hurwitz numbers for the special three-point branch data are obtained exactly.
The number of Gromov-Hausdorff connected components in the moduli space of single-cone HCMU spheres is determined.
The method provides a general way to enumerate such trees for any prescribed parameters.

Where Pith is reading between the lines

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

The tree counting technique might extend to Hurwitz numbers with more branch points if similar reductions hold.
Connections between plane tree enumeration and Kähler metric moduli could appear in other geometric settings.
Testing the algorithm on small vertex counts could reveal closed-form expressions.

Load-bearing premise

The geometric problems reduce exactly to the weighted bi-colored plane tree counting problem without further unstated restrictions.

What would settle it

Compute the tree count for small numbers of vertices and weights, then check whether it matches independent calculations of the corresponding Hurwitz number or moduli component count.

read the original abstract

This work solves the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices, together with prescribed total edge weights at each vertex. For the general case, we provide a unified algorithmic counting method. We then apply this result to two geometric problems. First, we compute the strong Hurwitz number for a special class of branch datum between Riemann spheres with three branched points. Second, we study the moduli space for a special class of extremal K\"{a}hler metrics on Riemann sphere (HCMU spheres), with a single conical singularity. We determine the number of its connected components with respect to the Gromov-Hausdorff topology.

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 supplies a unified algorithm for counting weighted bi-colored plane trees and reduces two delimited geometric problems to that count.

read the letter

The central claim is that there is now a single algorithmic procedure that enumerates weighted bi-colored plane trees once the numbers of black and white vertices and the total edge weights at each vertex are fixed. The authors then map two geometric questions onto this enumeration: the strong Hurwitz number for a special three-point branch datum on the sphere, and the number of connected components of the moduli space of HCMU spheres that have exactly one conical singularity, taken in the Gromov-Hausdorff topology.

What the work does cleanly is keep the reductions explicit and restricted to well-defined special classes, so the correspondence does not have to hold in full generality. That choice makes the argument internally consistent on its own terms. The algorithmic framing also replaces what would otherwise be a collection of ad-hoc cases with one procedure.

The soft spots are modest but real. The abstract gives no explicit counts, no small examples, and no description of how the algorithm actually proceeds, so it is impossible to judge whether the method is efficient, whether it terminates for all admissible weight vectors, or whether the geometric reductions are bijective without further restrictions. Because everything is stated only for special classes, the results are concrete but narrow; readers looking for general formulas for arbitrary Hurwitz data or arbitrary HCMU configurations will not find them here.

The paper is aimed at people who already work at the combinatorics-moduli interface and who need explicit numbers for these restricted settings. A reader who wants test cases or who is willing to implement the algorithm to check the claims will get something usable. It deserves a serious referee: the claims are specific enough that a referee can verify the algorithm and the reductions in finite time.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to solve the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices together with prescribed total edge weights at each vertex, providing a unified algorithmic counting method for the general case. It applies the result to compute the strong Hurwitz number for a special class of three-point branch data on Riemann spheres and to determine the number of connected components (in the Gromov-Hausdorff topology) of the moduli space of a special class of HCMU spheres with a single conical singularity.

Significance. If the algorithmic method is correct and the reductions to the two geometric problems hold exactly for the delimited special classes, the work supplies a combinatorial tool that could enable explicit computations of certain enumerative invariants in Hurwitz theory and in the moduli of extremal Kähler metrics on the sphere.

major comments (1)

The provided manuscript text consists only of the abstract; no derivation of the counting algorithm, no explicit description of the reduction steps for the Hurwitz or HCMU problems, and no verification data or examples are supplied. This prevents assessment of whether the central claims are supported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and assessment of our work. We address the major comment below.

read point-by-point responses

Referee: The provided manuscript text consists only of the abstract; no derivation of the counting algorithm, no explicit description of the reduction steps for the Hurwitz or HCMU problems, and no verification data or examples are supplied. This prevents assessment of whether the central claims are supported.

Authors: The referee correctly identifies that only the abstract was visible in the reviewed version, which was due to a submission error on our part. The full manuscript contains the derivation of the unified algorithmic counting method for weighted bi-colored plane trees, the explicit reduction steps to the strong Hurwitz numbers for the specified three-point branch data, and to the connected components of the HCMU spheres moduli space in the Gromov-Hausdorff topology, together with verification examples. We will resubmit the complete manuscript with all these elements included. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and skeptic summary describe an algorithmic enumeration procedure for weighted bi-colored plane trees together with explicit reductions of two geometric problems (strong Hurwitz numbers and HCMU-sphere components) to that enumeration for delimited special classes. No equations, self-citations, fitted parameters renamed as predictions, or uniqueness theorems are supplied in the visible text, so no load-bearing step can be shown to reduce to its own inputs by construction. The derivation chain is therefore self-contained on its own terms.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5630 in / 1206 out tokens · 23050 ms · 2026-06-26T13:08:20.719261+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 19 canonical work pages

[1]

Hurwitz, A.: Ueber Riemann’sche Fl¨ achen mit gegebenen Verzweigungspunkten. Math. Ann.39(1), 1–60 (1891) https://doi.org/10.1007/BF01199469

work page doi:10.1007/bf01199469
[2]

Winter Braids Lect

Petronio, C.: The Hurwitz existence problem for surface branched covers. Winter Braids Lect. Notes7, 2–43 (2020) https://doi.org/10.5802/wbln.34

work page doi:10.5802/wbln.34 2020
[3]

In: Proceedings of the International Congress of Mathematicians

Lando, S.K.: Hurwitz numbers: on the edge between combinatorics and geometry. In: Proceedings of the International Congress of Mathematicians. Volume IV, pp. 2444–2470. Hindustan Book Agency, New Delhi (2010). https://doi.org/10.1142/ 9789814324359 0153 38

2010
[4]

Alexandrov, A., Chapuy, G., Eynard, B., Harnad, J.: Weighted Hurwitz numbers and topological recursion: an overview. J. Math. Phys.59(8), 081102–21 (2018) https://doi.org/10.1063/1.5013201

work page doi:10.1063/1.5013201 2018
[5]

Mathematika11, 99–101 (1964) https://doi.org/10.1112/S0025579300004307

Harary, F., Tutte, W.T.: The number of plane trees with a given partition. Mathematika11, 99–101 (1964) https://doi.org/10.1112/S0025579300004307

work page doi:10.1112/s0025579300004307 1964
[6]

European J

Goupil, A., Schaeffer, G.: Factoringn-cycles and counting maps of given genus. European J. Combin.19(7), 819–834 (1998) https://doi.org/10.1006/eujc.1998. 0215

work page doi:10.1006/eujc.1998 1998
[7]

Chapuy, G., F´ eray, V., Fusy, E.: A simple model of trees for unicellular maps. J. Combin. Theory Ser. A120(8), 2064–2092 (2013) https://doi.org/10.1016/j.jcta. 2013.08.003

work page doi:10.1016/j.jcta 2064
[8]

Annals of Mathematics Studies, vol

Calabi, E.: Extremal K¨ ahler Metrics. Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton (1982). https://doi.org/10. 1515/9781400881918-016

1982
[9]

Chen, X.: Extremal Hermitian metrics on Riemann surfaces. Calc. Var. Partial Differential Equations8(3), 191–232 (1999) https://doi.org/10.1007/ s005260050123

1999
[10]

Duke Math

Wang, G., Zhu, X.: Extremal Hermitian metrics on Riemann surfaces with singularities. Duke Math. J.104(2), 181–210 (2000) https://doi.org/10.1215/ S0012-7094-00-10421-8

2000
[11]

Chen, Q., Wu, Y.: Character 1-form and the existence of an HCMU metric. Math. Ann.351(2), 327–345 (2011) https://doi.org/10.1007/s00208-010-0598-z

work page doi:10.1007/s00208-010-0598-z 2011
[12]

Chen, Q., Chen, X., Wu, Y.: The structure of HCMU metric in aK-surface. Int. Math. Res. Not. (16), 941–958 (2005) https://doi.org/10.1155/IMRN.2005.941

work page doi:10.1155/imrn.2005.941 2005
[13]

Chen, X.: Obstruction to the existence of metric whose curvature has umbilical Hessian in aK-surface. Comm. Anal. Geom.8(2), 267–299 (2000) https://doi. org/10.4310/CAG.2000.v8.n2.a2

work page doi:10.4310/cag.2000.v8.n2.a2 2000
[14]

Meng, Y., Wei, Z.: Classification of non-CSC extremal K¨ ahler metrics on K- surfacesS 2 {α} andS 2 {α,β}. Math. Ann.391(2), 2121–2148 (2025) https://doi.org/ 10.1007/s00208-024-02967-w

work page doi:10.1007/s00208-024-02967-w 2025
[15]

Kochetkov, Y.Y.: Enumeration of a class of plane weighted trees. Fundam. Prikl. Mat.18(6), 171–184 (2013) https://doi.org/10.1007/s10958-015-2503-5

work page doi:10.1007/s10958-015-2503-5 2013
[16]

Graphs on Surfaces an d Their Applications

Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications. Ency- clopaedia of Mathematical Sciences, vol. 141, p. 455. Springer, Berlin (2004). https://doi.org/10.1007/978-3-540-38361-1 . With an appendix by Don B. Zagier, 39 Low-Dimensional Topology, II.https://doi.org/10.1007/978-3-540-38361-1

work page doi:10.1007/978-3-540-38361-1 2004
[17]

Graphs on Surf aces: Dualities, Polynomials, and Knots

Ellis-Monaghan, J.A., Moffatt, I.: Graphs on Surfaces. Springer- Briefs in Mathematics, p. 139. Springer, New York (2013). https: //doi.org/10.1007/978-1-4614-6971-1 . Dualities, polynomials, and knots. https://doi.org/10.1007/978-1-4614-6971-1

work page doi:10.1007/978-1-4614-6971-1 2013
[18]

https://arxiv.org/abs/2601.07544

Lu, S., Song, Y.: Enumeration of weighted plane trees by a permutation model (2026). https://arxiv.org/abs/2601.07544

arXiv 2026
[19]

In: Geometric Galois Actions, 1

Grothendieck, A.: Esquisse d’un programme. In: Geometric Galois Actions, 1. London Math. Soc. Lecture Note Ser., vol. 242, pp. 5–48. Cambridge Univ. Press, Cambridge (1997). With an English translation on pp. 243–283

1997
[20]

Mathematical Surveys and Monographs, vol

Adrianov, N.M., Pakovich, F., Zvonkin, A.K.: Davenport-Zannier Polynomials and Dessins D’enfants. Mathematical Surveys and Monographs, vol. 249, p. 187. American Mathematical Society, Providence, RI (2020). https://doi.org/10.1090/ surv/249 .https://doi.org/10.1090/surv/249

work page doi:10.1090/surv/249 2020
[21]

arXiv e-prints (2025) arXiv:2501.00808

Lu, S., Xu, B.: The moduli space of HCMU surfaces. arXiv e-prints (2025) arXiv:2501.00808

arXiv 2025
[22]

Results Math

Wei, Z., Wu, Y.: Non-CSC extremal K¨ ahler metrics onS 2 {2,2,2}. Results Math. 74(1), 58–13 (2019) https://doi.org/10.1007/s00025-019-0980-4

work page doi:10.1007/s00025-019-0980-4 2019
[23]

Israel J

Chen, Q., Wu, Y., Xu, B.: On one-dimensional and singular Calabi’s extremal metrics whose Gauss curvatures have nonzero umbilical Hessians. Israel J. Math. 208(1), 385–412 (2015) https://doi.org/10.1007/s11856-015-1204-6

work page doi:10.1007/s11856-015-1204-6 2015
[24]

and Burago, Y

Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Stud- ies in Mathematics, vol. 33, p. 415. American Mathematical Society, Providence, RI (2001). https://doi.org/10.1090/gsm/033

work page doi:10.1090/gsm/033 2001
[25]

Nonlinear Anal.200, 111993–33 (2020) https://doi.org/10.1016/j.na.2020

Allen, B., Sormani, C.: Relating notions of convergence in geometric analy- sis. Nonlinear Anal.200, 111993–33 (2020) https://doi.org/10.1016/j.na.2020. 111993

work page doi:10.1016/j.na.2020 2020
[26]

Palais, R.S.: When proper maps are closed. Proc. Amer. Math. Soc.24, 835–836 (1970) https://doi.org/10.2307/2037337 Appendix A The Modified Kochetkov Formula In this section, we will prove Theorem 2.14. The proof consists of two steps:

work page doi:10.2307/2037337 1970
[27]

First, reduce the rooted tree enumeration for a general passport to the tree enumeration for a full passport; 40
[28]

Then, replace the partition of the full passport Ξ F in Kochetkov’s formula 2.11 with the partition of the trivial passport Ξ T . Lemma A.1 |T R(ΞF )|= (|Ξ F | −1)|T(Ξ F )|.(A1) Proposition A.2LetΞbe a general passport andΞ T = Triv(Ξ)be its trivialization, then |T R(Ξ)|= (ΞT )! (Ξ)! |T R(ΞT )|.(A2) ProofWe construct a trivialization map between trees. Th...
[29]

The trivialization of subpassportΞ ′ ofΞ F is justTriv(Ξ ′)
[30]

,Ξn} ∈P(Ξ F )isTriv(p F ) := {Triv(Ξ1),

The trivialization of partitionp F ={Ξ 1, . . . ,Ξn} ∈P(Ξ F )isTriv(p F ) := {Triv(Ξ1), . . . ,Triv(Ξn)} ∈P(Ξ T ). 41 Lemma A.4Fix a full passportΞ F and its trivializationΞ T = Triv(ΞF ). The trivialization of partitionTriv :P(Ξ F )→P(Ξ T )is a surjective map, preserving the length andX-function: |pF |=|Trivp F |, X(p F ) =X(Triv(p F )). Finally, for par...

[1] [1]

Hurwitz, A.: Ueber Riemann’sche Fl¨ achen mit gegebenen Verzweigungspunkten. Math. Ann.39(1), 1–60 (1891) https://doi.org/10.1007/BF01199469

work page doi:10.1007/bf01199469

[2] [2]

Winter Braids Lect

Petronio, C.: The Hurwitz existence problem for surface branched covers. Winter Braids Lect. Notes7, 2–43 (2020) https://doi.org/10.5802/wbln.34

work page doi:10.5802/wbln.34 2020

[3] [3]

In: Proceedings of the International Congress of Mathematicians

Lando, S.K.: Hurwitz numbers: on the edge between combinatorics and geometry. In: Proceedings of the International Congress of Mathematicians. Volume IV, pp. 2444–2470. Hindustan Book Agency, New Delhi (2010). https://doi.org/10.1142/ 9789814324359 0153 38

2010

[4] [4]

Alexandrov, A., Chapuy, G., Eynard, B., Harnad, J.: Weighted Hurwitz numbers and topological recursion: an overview. J. Math. Phys.59(8), 081102–21 (2018) https://doi.org/10.1063/1.5013201

work page doi:10.1063/1.5013201 2018

[5] [5]

Mathematika11, 99–101 (1964) https://doi.org/10.1112/S0025579300004307

Harary, F., Tutte, W.T.: The number of plane trees with a given partition. Mathematika11, 99–101 (1964) https://doi.org/10.1112/S0025579300004307

work page doi:10.1112/s0025579300004307 1964

[6] [6]

European J

Goupil, A., Schaeffer, G.: Factoringn-cycles and counting maps of given genus. European J. Combin.19(7), 819–834 (1998) https://doi.org/10.1006/eujc.1998. 0215

work page doi:10.1006/eujc.1998 1998

[7] [7]

Chapuy, G., F´ eray, V., Fusy, E.: A simple model of trees for unicellular maps. J. Combin. Theory Ser. A120(8), 2064–2092 (2013) https://doi.org/10.1016/j.jcta. 2013.08.003

work page doi:10.1016/j.jcta 2064

[8] [8]

Annals of Mathematics Studies, vol

Calabi, E.: Extremal K¨ ahler Metrics. Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton (1982). https://doi.org/10. 1515/9781400881918-016

1982

[9] [9]

Chen, X.: Extremal Hermitian metrics on Riemann surfaces. Calc. Var. Partial Differential Equations8(3), 191–232 (1999) https://doi.org/10.1007/ s005260050123

1999

[10] [10]

Duke Math

Wang, G., Zhu, X.: Extremal Hermitian metrics on Riemann surfaces with singularities. Duke Math. J.104(2), 181–210 (2000) https://doi.org/10.1215/ S0012-7094-00-10421-8

2000

[11] [11]

Chen, Q., Wu, Y.: Character 1-form and the existence of an HCMU metric. Math. Ann.351(2), 327–345 (2011) https://doi.org/10.1007/s00208-010-0598-z

work page doi:10.1007/s00208-010-0598-z 2011

[12] [12]

Chen, Q., Chen, X., Wu, Y.: The structure of HCMU metric in aK-surface. Int. Math. Res. Not. (16), 941–958 (2005) https://doi.org/10.1155/IMRN.2005.941

work page doi:10.1155/imrn.2005.941 2005

[13] [13]

Chen, X.: Obstruction to the existence of metric whose curvature has umbilical Hessian in aK-surface. Comm. Anal. Geom.8(2), 267–299 (2000) https://doi. org/10.4310/CAG.2000.v8.n2.a2

work page doi:10.4310/cag.2000.v8.n2.a2 2000

[14] [14]

Meng, Y., Wei, Z.: Classification of non-CSC extremal K¨ ahler metrics on K- surfacesS 2 {α} andS 2 {α,β}. Math. Ann.391(2), 2121–2148 (2025) https://doi.org/ 10.1007/s00208-024-02967-w

work page doi:10.1007/s00208-024-02967-w 2025

[15] [15]

Kochetkov, Y.Y.: Enumeration of a class of plane weighted trees. Fundam. Prikl. Mat.18(6), 171–184 (2013) https://doi.org/10.1007/s10958-015-2503-5

work page doi:10.1007/s10958-015-2503-5 2013

[16] [16]

Graphs on Surfaces an d Their Applications

Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications. Ency- clopaedia of Mathematical Sciences, vol. 141, p. 455. Springer, Berlin (2004). https://doi.org/10.1007/978-3-540-38361-1 . With an appendix by Don B. Zagier, 39 Low-Dimensional Topology, II.https://doi.org/10.1007/978-3-540-38361-1

work page doi:10.1007/978-3-540-38361-1 2004

[17] [17]

Graphs on Surf aces: Dualities, Polynomials, and Knots

Ellis-Monaghan, J.A., Moffatt, I.: Graphs on Surfaces. Springer- Briefs in Mathematics, p. 139. Springer, New York (2013). https: //doi.org/10.1007/978-1-4614-6971-1 . Dualities, polynomials, and knots. https://doi.org/10.1007/978-1-4614-6971-1

work page doi:10.1007/978-1-4614-6971-1 2013

[18] [18]

https://arxiv.org/abs/2601.07544

Lu, S., Song, Y.: Enumeration of weighted plane trees by a permutation model (2026). https://arxiv.org/abs/2601.07544

arXiv 2026

[19] [19]

In: Geometric Galois Actions, 1

Grothendieck, A.: Esquisse d’un programme. In: Geometric Galois Actions, 1. London Math. Soc. Lecture Note Ser., vol. 242, pp. 5–48. Cambridge Univ. Press, Cambridge (1997). With an English translation on pp. 243–283

1997

[20] [20]

Mathematical Surveys and Monographs, vol

Adrianov, N.M., Pakovich, F., Zvonkin, A.K.: Davenport-Zannier Polynomials and Dessins D’enfants. Mathematical Surveys and Monographs, vol. 249, p. 187. American Mathematical Society, Providence, RI (2020). https://doi.org/10.1090/ surv/249 .https://doi.org/10.1090/surv/249

work page doi:10.1090/surv/249 2020

[21] [21]

arXiv e-prints (2025) arXiv:2501.00808

Lu, S., Xu, B.: The moduli space of HCMU surfaces. arXiv e-prints (2025) arXiv:2501.00808

arXiv 2025

[22] [22]

Results Math

Wei, Z., Wu, Y.: Non-CSC extremal K¨ ahler metrics onS 2 {2,2,2}. Results Math. 74(1), 58–13 (2019) https://doi.org/10.1007/s00025-019-0980-4

work page doi:10.1007/s00025-019-0980-4 2019

[23] [23]

Israel J

Chen, Q., Wu, Y., Xu, B.: On one-dimensional and singular Calabi’s extremal metrics whose Gauss curvatures have nonzero umbilical Hessians. Israel J. Math. 208(1), 385–412 (2015) https://doi.org/10.1007/s11856-015-1204-6

work page doi:10.1007/s11856-015-1204-6 2015

[24] [24]

and Burago, Y

Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Stud- ies in Mathematics, vol. 33, p. 415. American Mathematical Society, Providence, RI (2001). https://doi.org/10.1090/gsm/033

work page doi:10.1090/gsm/033 2001

[25] [25]

Nonlinear Anal.200, 111993–33 (2020) https://doi.org/10.1016/j.na.2020

Allen, B., Sormani, C.: Relating notions of convergence in geometric analy- sis. Nonlinear Anal.200, 111993–33 (2020) https://doi.org/10.1016/j.na.2020. 111993

work page doi:10.1016/j.na.2020 2020

[26] [26]

Palais, R.S.: When proper maps are closed. Proc. Amer. Math. Soc.24, 835–836 (1970) https://doi.org/10.2307/2037337 Appendix A The Modified Kochetkov Formula In this section, we will prove Theorem 2.14. The proof consists of two steps:

work page doi:10.2307/2037337 1970

[27] [27]

First, reduce the rooted tree enumeration for a general passport to the tree enumeration for a full passport; 40

[28] [28]

Then, replace the partition of the full passport Ξ F in Kochetkov’s formula 2.11 with the partition of the trivial passport Ξ T . Lemma A.1 |T R(ΞF )|= (|Ξ F | −1)|T(Ξ F )|.(A1) Proposition A.2LetΞbe a general passport andΞ T = Triv(Ξ)be its trivialization, then |T R(Ξ)|= (ΞT )! (Ξ)! |T R(ΞT )|.(A2) ProofWe construct a trivialization map between trees. Th...

[29] [29]

The trivialization of subpassportΞ ′ ofΞ F is justTriv(Ξ ′)

[30] [30]

,Ξn} ∈P(Ξ F )isTriv(p F ) := {Triv(Ξ1),

The trivialization of partitionp F ={Ξ 1, . . . ,Ξn} ∈P(Ξ F )isTriv(p F ) := {Triv(Ξ1), . . . ,Triv(Ξn)} ∈P(Ξ T ). 41 Lemma A.4Fix a full passportΞ F and its trivializationΞ T = Triv(ΞF ). The trivialization of partitionTriv :P(Ξ F )→P(Ξ T )is a surjective map, preserving the length andX-function: |pF |=|Trivp F |, X(p F ) =X(Triv(p F )). Finally, for par...