Probabilistic enumeration and equivalence of nonisomorphic trees

TL;DR

The work delivers a new probabilistic proof of Otter's asymptotic for unlabelled trees, establishing $f_n \sim c_F n^{-5/2}\rho^{-n}$ with $c_F = 2\pi c_A^3$, and identifies a sharp total-variation equivalence between random Pólya trees and unlabelled free trees: $d_{\mathrm{TV}}(\mathrm{F}(\mathcal{A}_n), \mathsf{F}_n) \to 0$ as $n\to\infty$, with explicit bounds. The approach centers on a generating-function framework built from symmetry/cycle-index structures, yielding coefficient asymptotics via probabilistic arguments involving random variables $X$, $N$, and $S_N$ without relying on the dissymmetry equation. A key contribution is the rigorous transfer principle that allows properties of rooted, labeled objects to be transferred to unlabelled free objects, with precise total-variation control. The results extend to degree-restricted trees and to subcritical, unlabelled graph classes, using cycle-index methods and branching-process concentration to obtain analogous approximation theorems, thereby unifying transfer phenomena across tree-like combinatorial families.

Abstract

We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between random Pólya trees and random unlabelled trees tends to zero when the number of vertices tends to infinity. In order to demonstrate that our approach is not restricted to trees we extend our results to tree-like classes of graphs.

Figures (1)

• Figure 1: Rooted unlabelled trees with five vertices grouped according to the underlying unrooted tree.

Theorems & Definitions (7)

• Theorem 1.1: Otter MR0025715
• Theorem 1.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3: Medium deviation inequality for one-dimensional random walk