# A bijection of plane increasing trees with relaxed binary trees of right height at most one

## TL;DR

This paper establishes a bijection between plane increasing trees and relaxed binary trees of right height at most one, revealing new combinatorial insights and connections among various subclasses.

## Contribution

It introduces a novel bijection between two classes of trees and explores their parameter relationships, including limit theorems and subclass enumerations.

## Key findings

- Established a bijection between plane increasing trees and relaxed binary trees
- Proved central limit theorems for leaf statistics
- Connected over 20 subclasses to known counting sequences

## Abstract

Plane increasing trees are rooted labeled trees embedded into the plane such that the sequence of labels is increasing on any branch starting at the root. Relaxed binary trees are a subclass of unlabeled directed acyclic graphs. We construct a bijection between these two combinatorial objects and study the therefrom arising connections of certain parameters. Furthermore, we show central limit theorems for two statistics on leaves. We end the study by considering more than 20 subclasses and their bijective counterparts. Many of these subclasses are enumerated by known counting sequences, and thus enrich their combinatorial interpretation.