Nonleaf Patterns in Trees: Protected Nodes and Fine Numbers

TL;DR

This paper derives a formula for counting pattern matches in ordered trees, extending previous enumeration methods to include protected nodes and subtrees, with implications for tree structure analysis.

Contribution

It introduces a closed-form formula for pattern occurrences in ordered trees, incorporating protected nodes and subtree components, advancing tree-pattern enumeration techniques.

Findings

Derived a closed-form formula for pattern matches in trees

Provided enumeration of trees by protected and unprotected nodes

Extended previous tree-pattern enumeration methods

Abstract

A closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also contain components that match subtrees or sequences of subtrees. This result extends previous versatile tree-pattern enumeration formulae to incorporate components that are only allowed to match nonleaf subtrees and provides enumerations of trees by the number of protected (shortest outgoing path has two or more edges) or unprotected nodes.