# A recursive algorithm for trees and forests

## TL;DR

This paper introduces recurrence relations for counting rooted forests with varying numbers of roots, enabling derivation of classical formulas for different types of trees.

## Contribution

It presents a recursive approach to relate counts of rooted forests with k and k+1 roots, simplifying derivation of known tree enumeration formulas.

## Key findings

- Derived recurrence relations for rooted forests with k and k+1 roots.
- Unified framework for counting various types of trees.
- Simplified derivation of classical tree enumeration formulas.

## Abstract

Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.