### Abstract

Let Y_{k} (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z (ω) := ∑_{k≥0} Y_{k}(ω) is the total progeny of w. In this paper, we will prove various statistical properties of Z and Y_{k}. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that y_{k}(ω) := ∑ _{j=0}^{k} Y_{j} (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree w. We then proceed to study the joint probability distribution of Z and {Y_{k}}_{k}, and show that, as n → ∞, {Y_{k}/n}_{k} is asymptotically Gaussian under the conditional distribution P(· Z = n).

Original language | English |
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Pages (from-to) | 229-264 |

Number of pages | 36 |

Journal | Advances in Applied Probability |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 Mar 1 |

Externally published | Yes |

### Keywords

- Central limit theorem
- Galton-Watson tree
- Lagrange inversion
- Total progeny

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics