Distinct Fringe Subtrees in Random Trees

TL;DR

This paper investigates the number of distinct fringe subtrees in various random tree models, providing asymptotic estimates for their counts as the number of vertices grows large.

Contribution

It establishes the order of magnitude of the number of distinct fringe subtrees in two main classes of random trees, extending understanding of their structural complexity.

Findings

Number of distinct fringe subtrees in simply generated trees is on the order of n/√log n.

Number of distinct fringe subtrees in increasing trees is on the order of n/log n.

Provides asymptotic estimates under mild assumptions on 'distinct' fringe subtrees.

Abstract

A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in two types of random trees: simply generated trees and families of increasing trees (recursive trees, $d$-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what `distinct' means) in random trees with $n$ vertices is $n/\sqrt{\log n}$ for simply generated trees and $n/\log n$ for increasing trees.