Convergence and limits of finite trees

TL;DR

This paper explores the convergence of finite trees to a unique limit object called a dendron, extending concepts from graph limits to tree structures using sampling in normalized distance.

Contribution

It introduces dendrons as a new limit object for finite trees and proves their uniqueness, advancing the understanding of tree convergence in graph theory.

Findings

Finite trees converge to dendrons under sampling in normalized distance.

Dendrons are characterized as the exact limits of finite trees.

The limit dendron for a sequence of finite trees is unique.

Abstract

Motivated by the work of Lov\'asz and Szegedy on the convergence and limits of dense graph sequences, we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. Based on separable real trees, we introduce the notion of a dendron and show that the limits of finite trees are exactly the dendrons. We also prove that the limit dendron is unique.