A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. v{C}erny conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most
$(n-1)^2$. We introduce the notion of aperiodically $1-$contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the v{C}erny conjecture holds for aperiodically $1-$contracting automata. As a special case, we prove some results for circular automata.
We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more type
s and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.
