Action graphs and Catalan numbers

TL;DR

This paper introduces a sequence of directed graphs whose edge counts at each step correspond to Catalan numbers and establishes an isomorphism with planar rooted trees, revealing combinatorial structures.

Contribution

It defines a new class of directed graphs and proves their edge counts match Catalan numbers, linking graph theory with combinatorial tree structures.

Findings

Number of edges added at step k equals the kth Catalan number

Edges at step k are isomorphic to planar rooted trees with k edges

Establishes a combinatorial correspondence between graphs and trees

Abstract

We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step $k$ and the set of planar rooted trees with $k$ edges.