### Abstract

*t*

_{n})

_{n≥1}, and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees (

*t*

_{n})

_{n≥1}, the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of (

*t*

_{n})

_{n≥1}: a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).

Original language | English |
---|---|

Pages (from-to) | 15–58 |

Number of pages | 44 |

Journal | Random Structures and Algorithms |

Volume | 53 |

Issue number | 1 |

Early online date | 11 Jan 2018 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

### Fingerprint

### Keywords

- and/or trees
- local limit
- random Boolean functions
- random trees

### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*53*(1), 15–58. https://doi.org/10.1002/rsa.20758

**And/or trees : a local limit point of view.** / Broutin, Nicolas; Mailler, Cecile.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 53, no. 1, pp. 15–58. https://doi.org/10.1002/rsa.20758

}

TY - JOUR

T1 - And/or trees

T2 - a local limit point of view

AU - Broutin, Nicolas

AU - Mailler, Cecile

PY - 2018/8/1

Y1 - 2018/8/1

N2 - We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say (tn)n≥1, and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees (tn)n≥1, the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of (tn)n≥1 : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).

AB - We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say (tn)n≥1, and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees (tn)n≥1, the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of (tn)n≥1 : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).

KW - and/or trees

KW - local limit

KW - random Boolean functions

KW - random trees

UR - http://www.scopus.com/inward/record.url?scp=85040592199&partnerID=8YFLogxK

U2 - 10.1002/rsa.20758

DO - 10.1002/rsa.20758

M3 - Article

VL - 53

SP - 15

EP - 58

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -