An Efficient Scheme for the Generation of Ordered Trees in Constant Amortized Time

TL;DR

This paper introduces an algebraic scheme for generating ordered trees with n vertices efficiently, achieving constant average time per tree, which benefits combinatorial optimization and related fields.

Contribution

The paper presents a novel algebraic method to generate ordered trees with linear space and constant average time per tree, improving efficiency over previous approaches.

Findings

Generation of ordered trees in constant average time

Uses O(n) space for generating trees

Applicable to various combinatorial structures

Abstract

Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. In combinatorial optimization, generating ordered trees is relevant to evaluate candidate combinatorial objects. In this paper, we present an algebraic scheme to generate ordered trees with $n$ vertices with utmost efficiency; whereby our approach uses $\mathcal{O}(n)$ space and $\mathcal{O}(1)$ time in average per tree. Our computational studies have shown the feasibility and efficiency to generate ordered trees in constant time in average, in about one tenth of a millisecond per ordered tree. Due to the 1-1 bijective nature to other combinatorial classes, our approach is favorable to study the generation of binary trees with $n$ external nodes, trees with $n$ nodes, legal sequences of $n$ pairs of parentheses, triangulated $n$-gons, gambler's sequences and lattice paths. We believe our scheme may find its use in devising algorithms for planning and combinatorial optimization involving Catalan numbers.