# Entropy Bounds for Grammar-Based Tree Compressors

## TL;DR

This paper extends the concept of empirical entropy to binary trees and demonstrates that grammar-based tree compression can achieve encoding sizes close to this entropy measure, generalizing previous string compression results.

## Contribution

It introduces a new entropy measure for trees and shows that grammar-based tree encodings can be bounded by this measure, extending string compression theories to trees.

## Key findings

- Tree entropy bounds are established for grammar-based tree compressors.
- Binary encodings of trees are shown to be near the entropy limit.
- Generalization of string compression results to tree structures.

## Abstract

The definition of $k^{th}$-order empirical entropy of strings is extended to node labelled binary trees. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the $k^{th}$-order empirical entropy plus some lower order terms. This generalizes recent results for grammar-based string compression to grammar-based tree compression.