One-Parameter Leaf-Based Growth Model

• The paper presents a stochastic tree-growth model where attachment probabilities depend on leaf degree offset by a single parameter, a > 0.
• It employs master equations and generating functions to derive stationary leaf-degree distributions and reveals power-law, stretched-exponential, and exponential regimes.
• The model extends classic preferential attachment by incorporating leaf properties, offering insights for applications in phylogenetics, geometric statistics, and complex network analysis.

The one-parameter leaf-based growth model refers to a family of stochastic tree-growth processes where the probability of attaching a new vertex (or lineage, with domain-dependent terminology) depends on a single real parameter and a leaf-specific property, such as the leaf degree, age, or geometric descriptor. This article gives a rigorous account of such models as described in Hartle & Krapivsky (Hartle et al., 6 Nov 2025), Huckemann (Huckemann, 2010), and related age-dependent branching models (Keller-Schmidt et al., 2010). The principal focus is on the Hartle–Krapivsky model, which formalizes attachment preferences according to leaf degree offset by a parameter $a>0$, but closely related forms also arise in geometric shape statistics and phylogenetic modeling.

1. Leaf-Based Growth: Model Definition

In the canonical one-parameter leaf-based growth model for trees (Hartle et al., 6 Nov 2025), consider a tree with $N$ labeled vertices. Each vertex $i$ is assigned a leaf degree $\ell_i$, denoting the number of its neighbors that are leaves (i.e., vertices of degree one). The model evolves from a single-vertex tree ($N=1$) by iteratively adding a new vertex and connecting it via a single edge to an existing vertex. The probability that vertex $i$ is chosen as the attachment target is proportional to $\ell_i + a$, where $a > 0$ is the model parameter.

The normalization constant at time $N$ is

$$K(N) = \sum_{\ell \geq 0} (\ell + a) M_\ell(N) = N_1(N) + aN,$$

where $M_\ell(N)$ is the number of vertices of leaf degree $\ell$, and $N_1(N)$ is the total number of leaves. In continuous time, the equivalent is a vertex $i$ spawning a new edge at rate $\ell_i + a$.

This mechanism privileges connectivity to leaves rather than ordinary vertex degree, fundamentally altering the structural statistics of the resulting trees relative to classic preferential attachment ($\mathrm{PA}(\delta)$) models.

2. Master Equations and Generating-Function Approach

Let $M_\ell(N)$ denote the (random) count of vertices of leaf degree $\ell$ after $N$ growth steps. Its mean satisfies the recurrence: $\frac{d\,\E\,M_\ell}{dN} =\frac{(\ell-1+a)\,\E\,M_{\ell-1} + a(\ell+1)\,\E\,M_{\ell+1} - [(\ell+a)+a]\,\E\,M_\ell}{N_1+a\,N} + \delta_{\ell,0},$ with the $\delta_{\ell,0}$ tracing new arrivals as leaves of degree zero.

Assuming self-averaging ($\E M_\ell(N) \sim N m_\ell$), and denoting the leaf fraction $n_1 = \lim N_1/N$, the stationary distribution $m_\ell$ satisfies

$$[\ell(1+a) + 2a+n_1] m_\ell = (\ell-1+a) m_{\ell-1} + a(\ell+1) m_{\ell+1} + (a+n_1) \delta_{\ell,0}.$$

The leaf fraction is determined by solving

$$\frac{dN_1}{dN} = 1 - \frac{N_1}{N_1+aN} \quad \Longrightarrow \quad n_1(a) = \frac{1}{2}-a+\sqrt{a^2+\tfrac{1}{4}}.$$

The stationary distribution is most tractably extracted using generating functions: $m(z) = \sum_{\ell \ge 0} m_\ell z^\ell,$ which solves the ordinary differential equation

$$(1-z)(a-z) \frac{dm}{dz} - [\,a(1-z)+(a+n_1)\,] m(z) + (a+n_1) = 0,$$

subject to $m(1)=1$. For $a > 1$ explicit integral solutions in terms of hypergeometric and incomplete Beta functions are available.

3. Stationary Leaf-Degree Distributions and Tail Regimes

The closed-form and large-$\ell$ asymptotics of $m_\ell$ reveal three distinct regimes dependent on $a$.

Power-Law Regime ($0

$$m_\ell \sim C(a)\, \ell^{-\lambda(a)} ,\quad \lambda(a) = \frac{1+n_1(a)}{1-a}$$

with $C(a)$ a model-specific amplitude. $\lambda(a)$ varies smoothly from 2 at $a\to0$ to $\infty$ as $a \to 1^-$, with progressively heavier tails for smaller $a$.

Critical/Tricritical Regime ($a=1$)

$$m_\ell \sim D\, \ell^{-1/4} \exp[-2\sqrt{g \ell}] ,\quad g = \tfrac{\sqrt{5}+1}{2},\quad D\approx 5.3$$

The distribution exhibits a stretched-exponential cutoff with significant fluctuations up to $\ell \sim (\ln N)^2$.

Exponential Regime ($a>1$)

$$m_\ell \sim E(a)\, \ell^{-\lambda(a)} a^{-\ell} ,\quad E(a) = \frac{(a-1)^a}{\Gamma(a)}$$

The exponential term dominates, with algebraic corrections from $\lambda(a)$.

As $a\to\infty$ (random-recursive-tree limit), the leaf-degree distribution is approximately

$$m_\ell = \int_0^1 e^{-t} \frac{t^\ell}{\ell!}\,dt = \frac{\gamma(\ell+1,1)}{\Gamma(\ell+1)} ,$$

where $\gamma$ is the lower incomplete gamma function.

4. Comparative Analysis: Degree–Leaf-Degree Exponent Equivalence

Numerical simulations indicate that in the range $0$\mathrm{PA}(\delta)$ model, the degree exponent is $\gamma = 3 + \delta$. By analogy, in leaf-based PA, the scale-free regime aligns the exponent $\lambda(a)$ in both degree and leaf-degree distributions.

A plausible implication is that leaf-based PA furnishes a more flexible model than standard PA, interpolating continuously among power-law, stretched-exponential, and exponential degree statistics depending on setting of $a$. Notably, the stretched-exponential ($a=1$) and strictly exponential ($a > 1$) regimes are inaccessible in ordinary linear PA models.

5. Additional Characteristics and Extensions

Several ancillary features have been computed for the one-parameter model.

• Primordial-vertex Law: For $a>1$, the stationary leaf-degree distribution of the first vertex is

$$\pi_\ell = \frac{(a-1)^a}{a^{\ell+a}} \frac{\Gamma(\ell+a)}{\Gamma(a)\,\Gamma(\ell+1)},$$

matching the global exponential-algebraic tail structure.

• Age-Stratified Distribution: For each arrival index $j$, the probability for leaf-degree $\ell$ in large-$N$ trees scales according to $\Pi_\ell(x)$, with $x=j/N$ varying from primordial law ($x \to 0$) to extinction ($x \to 1$).
• Total Leaf Count Distributions: In the random-recursive limit, exact finite-$N$ formulae (in terms of Eulerian numbers) exist for the leaf-count distribution, including closed-form for cumulants and generating functions.
• Extensions: The leaf-based model admits generalization to cyclic graphs, leaf-based deletion, and empirical networks (phylogenies, metabolic pipelines, power grids). Empirically, sparse graphs often feature leaf dominance—a phenomenon underrepresented relative to classic degree statistics in theoretical network science.

6. Phenomenological Interpretation of the Offset Parameter

The parameter $a$ modulates the transition between distinct statistical regimes, analogous to the “initial attractiveness” $\delta$ in degree-based PA but defined on leaf-degree instead. Increasing $a$ from 0 to 1 yields progressively steeper distribution tails, culminating at $a=1$ in a tricritical point where power-law scaling ceases and stretched-exponentials emerge. For $a>1$, strong offset localizes structure exponentially.

A plausible implication is that leaf-based PA, with offset parameter $a$, constitutes a one-parameter universality class for tree growth, encompassing and extending behaviour observed in both classic preferential attachment and age-dependent branching, as well as in geometric and evolutionary models for leaf growth (Hartle et al., 6 Nov 2025, Keller-Schmidt et al., 2010). The distinct regimes—scale-free ($a<1$), stretched-exponential ($a=1$), and exponentially localized ($a>1$)—characterize a model spectrum that is not accessible under degree-based PA, suggesting new avenues for modeling boundary phenomena in real-world sparse graphs.

7. Related Models and Domains

Geometric shape statistics (Huckemann, 2010) apply one-parameter leaf-based modeling to planar leaf growth, wherein shape evolution along geodesic curves in Kendall’s complex shape space can be inferred and statistically characterized with strong consistency and CLTs. In age-dependent branching (Keller-Schmidt et al., 2010), a parameter $\alpha$ controls speciation as a function of tip age, yielding critical points and algebraic or logarithmic scaling in tree depth, which recapitulate similar tricritical behaviour seen in leaf-based PA formulations.

Tables summarizing the tail regimes:

Regime	Parameter Range	Leaf-Degree Distribution $m_\ell$
Power-law	$0	$C(a)\,\ell^{-\lambda(a)}$
Stretched-exp	$a=1$	$D\,\ell^{-1/4} \exp[-2\sqrt{g\,\ell}]$
Exponential	$a>1$	$E(a)\,\ell^{-\lambda(a)}\,a^{-\ell}$

The leaf-based approach is thus a robust and extensible framework for generating, analyzing, and interpreting tree structures where leaves play a disproportionate or functionally significant role.