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# Absorbing Patterns in BST-Like Expression-Trees

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## Acknowledgements

The authors would like to thank Arnaud Carayol and Cyril Nicaud for their very useful advice and remarks in the writing of this paper.

## Cite As

Florent Koechlin and Pablo Rotondo. Absorbing Patterns in BST-Like Expression-Trees. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.48

## Abstract

In this article we study the effect of simple semantic reductions on random BST-like expression-trees. Such random unary-binary expression-trees are often used in benchmarks for model-checking tools. We consider the reduction induced by an absorbing pattern for some given operator ⊛, which we apply bottom-up, producing an equivalent (and smaller) tree-expression. Our main result concerns the expected size of a random tree, of given input size n → ∞, after reduction. We show that there are two different thresholds, leading to a total of five regimes, ranging from no significant reduction at all, to almost complete reduction. These regimes are completely characterized according to the probability of the absorbing operator. Our results prove that random BST-like trees have to be considered with care, and that they offer a richer range of behaviours than uniform random trees.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Discrete mathematics
##### Keywords
• BST trees
• absorbing pattern
• reduction
• analytic combinatorics

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## References

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