A random recursive tree model with doubling events

TL;DR

This work introduces a random recursive tree modified by global doubling events at the root and analyzes its asymptotic geometry. Using a combination of moment methods and stochastic approximation, it proves linear growth in size, derives the limiting degree distribution identical to classical random recursive trees, and characterizes the height profile with a nontrivial random shift reflecting doubling events. A continuous-time embedding and renewal-style arguments underpin the height results, and a comparison with a all-nodes-doubling variant reveals superlinear growth in expectation. Overall, the paper illuminates how nonlocal growth mechanisms reshape global tree structure and height statistics compared to standard recursive trees.

Abstract

We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.

Figures (2)

• Figure 1: Steps in the construction of a random doubling tree. The root is drawn grey and, at each step, the randomly selected node is circled in red. In the first and fourth steps, the selected node is the root and a doubling event occurs. At the other steps, a non-root node is selected and a leaf is added to that node.
• Figure 2: One can see how the process $(N(t))_{t\geq 0}$ can be coupled with a Yule process $(Y(t))_{t\geq 0}$ so that, for all $t\geq 0$, $Y(t+\ell_{D(t)}) = N(t)$, where $D(t)$ is the number of doubling events before time $t$. The Yule process is the concatenation of the black and grey parts of the curve, whilst $(N(t))_{t\geq 0}$ is the curve obtained by only keeping the black parts and gluing them as if time-warps made us skip the intervals of time in grey. The two pairs of distances highlighted on the left-hand side are such that the two arrows in one pair have the same length: this means that the grey intervals are intervals during which the Yule process doubles in size. Note that, although both $Y$ and $N$ are jump processes that take value in $\mathbb N$, we have here represented them as continuous curves (with jumps for $N$ when it doubles); this is just for ease of representation. One can see that, if the total length of all the purple intervals is $t$, then, indeed, $D(t) = 2$ and $N(t)$, which is the value highlighted by large a purple dot, equals $Y(t+\ell_2)$, as claimed in the proof of Proposition \ref{['prop:height_LB']}.

Theorems & Definitions (30)

• Proposition 1.1: Asymptotic size
• Remark 1.2
• Theorem 1.3: Degree distribution
• Theorem 1.4: Height profile
• Proposition 1.5: Lower bound on the height
• Lemma 2.1
• proof
• Remark 2.2
• proof : Proof of Proposition \ref{['prop:cvBn']}
• Proposition 2.3