We consider Galton-Watson branching process with offspring , starting with 1 individual in generation 0. is a non-negative, integer-valued random variable with mean 1 and variance . We note the offspring distribution.

We write for the family tree of this branching process and the probability space of all trees with the law induced by . We note the space of size trees endowed by the conditional law given (under this condition, Galton-Watson trees are simple trees). The law on will be denoted by .

Let be a node of and note the number of ancestors of ( excluded) that have children. We note the depth of the node (that is ).

We ``show'' strong and uniform properties of the random variables :

**Theorem** *Let be a positive real number.*

*For any
*

That is, is roughly speaking proportional to , and this ``uniformly'' in .

To obtain this result (and other ones), we derive properties of ladder variables (the ladder variables are the random variables that intervene in the records of a random walk: time of the records, values of the record, increment that gives a record).

Application to moderate deviations of finite branching random walk are also given.

Please send comments and corrections to Thomas Klausner.