---
title: Scaled-Attachment Random Recursive Trees
url: https://www.emergentmind.com/topics/scaled-attachment-random-recursive-trees-sarrts
type: topic
---

# Scaled-Attachment Random Recursive Trees

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 [1210.7168] [1611.01306]. 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 [1611.01306].

## 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 [1210.7168].

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 [1210.7168].

## 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 [1611.01306].

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 [1210.7168][1611.01306].

## 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 [1611.01306]. 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" [1210.7168]
- Ross, Wen, "Scaling limits for some random trees constructed inhomogeneously" [1611.01306]

Source: https://www.emergentmind.com/topics/scaled-attachment-random-recursive-trees-sarrts