# Subtrees of a Given size in Schroeder Trees

## TL;DR

This paper derives a generating function for counting vertices with subtrees of size k in Schroeder trees and applies the method to Motzkin trees to compute related probabilities.

## Contribution

It introduces a technique to calculate subtree size probabilities in Schroeder and Motzkin trees using generating functions.

## Key findings

- Derived the generating function for subtree sizes in Schroeder trees
- Calculated probabilities of vertices having subtrees of size k
- Extended the method to Motzkin trees

## Abstract

In this paper we find the generating function for the number of vertices which have $k$ elements in their subtree and use this generating function to calculate the probability that a vertex has a size $k$ subtree. We also show how this same technique can be applied to calculate the probabilities for other trees and specifically apply it to Motzkin trees.