Scaled-Attachment Random Recursive Trees

• SARRTs are a class of random recursive tree models defined by nonuniform, random scaling attachment rules that generalize traditional recursive trees.
• They exhibit explicit logarithmic scaling for typical depth and height, with constants derived from renewal theory and large deviation techniques.
• The models extend to continuum limits where rescaling produces real trees, linking discrete structures to objects like Aldous's Brownian CRT.

Scaled-Attachment Random Recursive Trees (SARRTs) are a general class of random recursive tree models in which the location to which each new node attaches is determined by a random scaling process rather than uniform selection. This construction generalizes the classical random recursive tree (RRT) and admits a much broader array of depth and metric behaviors, encompassing nonuniform attachment rules, inhomogeneous tree growth, and rich continuum limits (Devroye et al., 2012, &&&1&&&). Key features include explicit logarithmic scaling constants for depth extremes and convergence to real trees under appropriate rescaling.

1. Discrete Model Definition and Attachment Rule

A SARRT is defined via a sequential growth process on vertices $V = \{0,1,2,\ldots,N\}$. For $i=1,\ldots,N$, node $i$ is attached to a parent node with label given by the rule

$\text{Parent}(i) := \lfloor i X_i \rfloor$

where $(X_0, X_1, \ldots, X_N)$ is a sequence of i.i.d.\ random variables with common distribution on $[0,1)$. The attachment process is Markovian but modulated by the random scaling inherent in the $X_i$.

This construction allows for non-uniform attachment: the attachment probability depends on the realization of $X_i$. The traditional RRT is recovered when $X_i \sim \mathrm{Unif}[0,1)$, yielding uniform attachment among available parents.

A related combinatorial model employs a parameter $\ell \in \mathbb{N}^+$. At each discrete time step, a vertex is added by splitting an edge chosen uniformly at random; every $\ell$ steps, a new leaf is appended. This yields a sequence of unlabeled rooted trees $T^{(\ell)}(n)$ with size growing in both vertices and leaves and provides a framework for rigorous scaling limits (Ross et al., 2016).

2. Asymptotic Depth and Height Parameters

The depth $D_i$ of node $i$ denotes the distance (in edges) from $i$ to the root. Three canonical depth parameters characterize the large $n$ regime:

• Typical depth $D_n$: depth of the last-inserted node.
• Tree height $H_n$: $\max_{1 \leq i \leq n} D_i$.
• Minimum depth among youngest half $M_n$: $\min_{n/2 \leq i \leq n} D_i$.

Their asymptotic behaviors are governed by logarithmic laws: $$D_n \sim \frac{1}{\mu} \log n, \qquad H_n \sim \alpha_{\mathrm{max}} \log n, \qquad M_n \sim \alpha_{\mathrm{min}} \log n$$ for explicit constants $\mu$, $\alpha_{\mathrm{max}}$, $\alpha_{\mathrm{min}}$ depending only on the law of $X_0$. These results hold whenever $X_0$ has a density, ensuring nondegeneracy of the asymptotics (Devroye et al., 2012).

The computation of these constants proceeds as follows. Set $Y = -\log X_0$ and denote its mean and variance by $\mu = \mathbb{E}[Y]$, $\sigma^2 = \operatorname{Var}(Y)$. Define the log-moment generating function $\Lambda(\lambda) = \log \mathbb{E}[e^{\lambda Y}]$ and its Cramér/Laplace transform $$\Lambda^*(z) = \sup_{\lambda \in \mathbb{R}}\{\lambda z - \Lambda(\lambda)\}$$, then

$\Psi(c) = c \Lambda^*(-1/c)$

The constants are then given by

$$\alpha_{\mathrm{max}} = \inf\{c > 1/\mu : \Psi(c) > 1\}, \qquad \alpha_{\mathrm{min}} = \sup\{0 \leq c < 1/\mu : \Psi(c) > 1\}$$

with $1/\mu < \alpha_{\mathrm{max}} < \infty$, $0 \leq \alpha_{\mathrm{min}} < 1/\mu$ when $X_0$ is nondegenerate.

3. Special Case: Uniform Distribution and Explicit Constants

For $X_0 \sim \mathrm{Unif}[0,1)$, one has $Y \sim \mathrm{Exp}(1)$, $\mu=1$ and $\sigma^2=1$. The Laplace and Cramér transforms simplify to

$$\Lambda(\lambda) = -\log(1-\lambda), \quad \Lambda^*(z) = z - 1 - \log z$$

yielding

$\Psi(c) = 1 - c - c \log c$

The equation $\Psi(c)=1$ is solved by $c=e$, so $\alpha_{\mathrm{max}}=e$ and thus the maximal tree height satisfies $H_n \sim e\log n$. Matching lower and upper bounds for height can be obtained without recourse to branching random walks, using Chernoff bounds, union bounds, and renewal-theoretic arguments (Devroye et al., 2012).

4. Scaling Limits and Continuum Real Trees

A continuum limit for SARRTs is established by embedding the discrete tree in a rescaled metric space and passing to the limit in the Gromov–Hausdorff–Prokhorov (GHP) topology. The limiting object is constructed via a line-breaking process on $\mathbb{R}^+$:

• Consider an inhomogeneous Poisson process on $(0,\infty)$ with rate $(\ell+1) t^\ell dt$, $\ell \geq 1$.
• Its jump times $C_0 = 0 < C_1 < C_2 < \cdots$ determine branch lengths.
• At each step, a new segment of length $C_{k+1} - C_k$ is attached at a point chosen uniformly with respect to length measure on the current tree.
• The projective limit yields a compact real tree $T$ with an intrinsic metric, and a canonical uniform leaf measure $\mu$ is inherited as the weak limit of uniform measures on leaves at each finite stage (Ross et al., 2016).

For $\ell=1$, the discrete process coincides with Rémy's algorithm and the continuum limit is Aldous's Brownian Continuum Random Tree (CRT), with Poisson line-breaking rate $2t dt$.

5. Detailed Proof Techniques and Renewal Theories

The depth and height results leverage several probabilistic tools:

• The evolution of labels along the ancestral line of a node can be linearized as $$\log L(n,j) \approx \log n - (Y_0 + Y_1 + \ldots + Y_{j-1})$$.
• The typical depth $D_n$ is then a hitting time for a sum of i.i.d.\ increments, facilitating the use of renewal theory and large deviation techniques.
• Chernoff bounds and union bounds are employed to obtain high-probability upper bounds for the tree height.
• Lower bounds on tree height (and, analogously, minimum depths) are derived via the second-moment (Chung–Erdős) argument and precise tail estimates from Cramér's theorem.
• In the continuum setting, couplings with Beta–Gamma and Dirichlet fragmentations justify the matching of discrete tree skeletons to the limit real tree.
• Coupling arguments with inhomogeneous Pólya urns control the number of discrete steps along arcs of the continuum tree, with sharp moment and maximal subtree size estimates (Devroye et al., 2012, Ross et al., 2016).

6. Generalizations, Special Cases, and Structural Phenomena

SARRTs encompass several natural and deterministic tree structures as special or limiting cases:

• If $X_0 \equiv \theta \in (0,1)$ almost surely, the process yields a deterministic $m$-ary complete tree with height $\approx (1/\mu)\log n$.
• Choosing $X_0 = \max\{U_1, \ldots, U_k\}$ or $X_0 = \min\{U_1, \ldots, U_k\}$, where $U_i$ are independent $\mathrm{Unif}[0,1)$, interpolates between "greedy" distance trees and uniform random DAGs, with explicit formulas for the scaling exponents.
• Power-of-choice models can be implemented by sampling multiple independent $X$ values and attaching to the minimizer, producing nontrivial effects on tree distances.

The choice of $X_0$ generates a spectrum of scaling constants, enabling richer phase-transition behavior for tree height and minimum depths than observed in the classical uniform RRT.

7. Connection to Rémy's Algorithm, CRTs, and Related Models

When $\ell=1$ in the edge-splitting formulation, the discrete process coincides precisely with Rémy's construction of uniform (leaf-labeled) binary trees. In this case, the scaling limit is the Brownian CRT constructed via Aldous's Poisson line-breaking process. The parameter $\ell$ can be interpreted as controlling the exponent of the attachment process in the continuum limit, with $\ell=1$ producing the CRT and higher $\ell$ yielding inhomogeneous generalizations (Ross et al., 2016). For $\ell = 0$, the limiting object is star-like and not generally included within the SARRT framework.

These connections highlight the role of SARRTs as a unifying framework for discrete and continuous models of random tree growth with broad applicability in probability, combinatorics, and statistical physics.

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Selected References:

• Devroye, Fawzi, Fraiman, "Depth properties of scaled attachment random recursive trees" (Devroye et al., 2012)
• Ross, Wen, "Scaling limits for some random trees constructed inhomogeneously" (Ross et al., 2016)