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How Many Stemmata with Root Degree k?

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how-many-stemmata-with-root-degree-k-2Probabilitymath.COmath.PRposed by Armin Hoenen, Steffen Eger, Ralf Gehrkerecorded: open · 1 machine check, unexamined

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Statement

From this, we (very strongly) conjecture that root bifurcating Greg trees have an asymptotic probability of above (and close to) 0.606.

Context

Candidate 2 of the open problems stated in "How Many Stemmata with Root Degree k?", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    Problem: Reconstructed conjecture: let gmg_m be the number of rooted Greg trees with mm labeled surviving nodes, where unlabeled nodes are allowed but must have at least two children. Let gm,2g_{m,2} be the number of such trees whose root has degree 22. The conjecture says

    R2(m):=gm,2gmR_2(m):=\frac{g_{m,2}}{g_m}

    has an asymptotic value above and close to 0.6060.606.

    Result: The conjecture is true. In fact,

    lim⁡m→∞R2(m)=e−1/2=0.6065306597….\lim_{m\to\infty}R_2(m)=e^{-1/2}=0.6065306597\ldots .

    Let

    G(z)=∑m≥1gmzmm!.G(z)=\sum_{m\ge1}g_m\frac{z^m}{m!}.

    A rooted Greg tree has either a labeled root with an unordered set of Greg subtrees, or an unlabeled root with an unordered set of at least two Greg subtrees. Hence

    G=zeG+(eG−1−G),G=ze^G+(e^G-1-G),

    equivalently

    (1+z)eG=1+2G.(1)(1+z)e^G=1+2G. \tag{1}

    If B(z)=∑m≥1gm,2zm/m!B(z)=\sum_{m\ge1}g_{m,2}z^m/m!, then root degree 22 gives

    B(z)=1+z2G(z)2.(2)B(z)=\frac{1+z}{2}G(z)^2. \tag{2}

    From (1),

    z=Φ(G),Φ(y)=(1+2y)e−y−1.z=\Phi(G),\qquad \Phi(y)=(1+2y)e^{-y}-1.

    The dominant critical point satisfies

    Φ′(y)=e−y(1−2y)=0,\Phi'(y)=e^{-y}(1-2y)=0,

    so

    τ=12,ρ=Φ(τ)=2e−1/2−1.\tau=\frac12,\qquad \rho=\Phi(\tau)=2e^{-1/2}-1.

    The standard smooth implicit-function schema applies: the defining function is analytic with nonnegative aperiodic coefficients, and

    Φ′′(τ)=−2e−1/2≠0.\Phi''(\tau)=-2e^{-1/2}\ne0.

    Thus

    G(z)=τ−c1−z/ρ+O(1−z/ρ)G(z)=\tau-c\sqrt{1-z/\rho}+O(1-z/\rho)

    in a Δ\Delta-domain at ρ\rho, with c>0c>0. Therefore

    [zm]G(z)∼c2πρ−mm−3/2.[z^m]G(z)\sim \frac{c}{2\sqrt\pi}\rho^{-m}m^{-3/2}.

    Using (1), rewrite (2) as

    B(z)=H(G(z)),H(y)=12y2(1+2y)e−y.B(z)=H(G(z)),\qquad H(y)=\frac12 y^2(1+2y)e^{-y}.

    Hence

    B(z)=H(τ)−cH′(τ)1−z/ρ+O(1−z/ρ),B(z)=H(\tau)-cH'(\tau)\sqrt{1-z/\rho}+O(1-z/\rho),

    and

    [zm]B(z)[zm]G(z)→H′(τ).\frac{[z^m]B(z)}{[z^m]G(z)}\to H'(\tau).

    Now

    H′(y)=12e−y(2y+5y2−2y3),H'(y)=\frac12e^{-y}(2y+5y^2-2y^3),

    so

    H′ ⁣(12)=e−1/2.H'\!\left(\frac12\right)=e^{-1/2}.

    Since the factor m!m! cancels between exponential coefficients,

    lim⁡m→∞gm,2gm=e−1/2.\lim_{m\to\infty}\frac{g_{m,2}}{g_m}=e^{-1/2}.

    This is indeed >0.606>0.606 and <0.607<0.607.

    Citation: The conjecture and definitions are from Hoenen, Eger, and Gehrke, “How Many Stemmata with Root Degree kk?”, MOL 2017. The analytic tool used is the smooth implicit-function schema and transfer theorem of Flajolet–Sedgewick, Analytic Combinatorics.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proof attacks the correct quantity R2(m)R_2(m) for rooted Greg trees. The species equations for all Greg trees and for root degree 22,

      G=zeG+(eG−1−G),B=1+z2G2,G=ze^G+(e^G-1-G),\qquad B=\frac{1+z}{2}G^2,

      are correct. The dominant square-root singularity occurs at G=1/2G=1/2, ρ=2e−1/2−1\rho=2e^{-1/2}-1, and standard transfer gives

      [zm]B[zm]G→e−1/2=0.606530659…>0.606.\frac{[z^m]B}{[z^m]G}\to e^{-1/2}=0.606530659\ldots>0.606.

      I found no prior explicit resolution of this limiting ratio in the checked sources; OEIS records the relevant sequences/EGF but not the asymptotic ratio.

      Novelty assessment

      TYPE1

      Classification rationale: The limit e−1/2e^{-1/2} appears genuinely not to be stated in the accessible literature, but the contribution is very small combinatorially: once the known EGF (1+z)eG=1+2G(1+z)e^{G}=1+2G is used, the root-degree-2 EGF and the coefficient ratio follow by a routine singularity-analysis calculation. This would be better as an OEIS note or short addendum to the stemmatology paper than as a standalone combinatorics-journal paper.

      Literature check: I found the original conjecture in Hoenen–Eger–Gehrke (MOL 2017), where only numerical evidence and the “above and close to 0.606” conjecture are given. OEIS A005264 records the EGF, Lambert-W form, and total asymptotic for rooted Greg trees; OEIS A286432 records the root-degree-2 sequence and finite formula, but not the limiting ratio. Related sources on Greg trees/phylogenetic trees, including Felsenstein, Foulds–Robinson, Flight, and Josuat-Vergès, do not appear to state this root-bifurcation limit. Searches of web results, GitHub, OEIS, and OpenAlex-visible metadata did not reveal a prior explicit e−1/2e^{-1/2} resolution.

      Citation: Hoenen, A.; Eger, S.; Gehrke, R. “How Many Stemmata with Root Degree kk?” Proceedings of MOL 2017, 11–21. See also OEIS A005264 and A286432; Flajolet–Sedgewick, Analytic Combinatorics, for the standard singularity-transfer machinery.

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