Postorder Preimages

TL;DR

This paper develops a combinatorial and geometric framework to count specific decreasing plane trees with a given postorder permutation, with applications to stack-sorting algorithms.

Contribution

It introduces a novel method for counting trees with a specified postorder in certain sets, extending understanding of tree permutations and sorting algorithms.

Findings

Provides a counting method for trees with a given postorder permutation.

Applies the method to analyze the deterministic stack-sorting algorithm.

Offers new insights into the structure of decreasing plane trees.

Abstract

Given a set $Y$ of decreasing plane trees and a permutation $\pi$, how many trees in $Y$ have $\pi$ as their postorder? Using combinatorial and geometric constructions, we provide a method for answering this question for certain sets $Y$ and all permutations $\pi$. We then provide applications of our results to the study of the deterministic stack-sorting algorithm.