  The frog model is a growing system of random walks where a particle is added
whenever a new site is visited. A longstanding open question is how often the
root is visited on the infinite $d$-ary tree. We prove the model undergoes a
phase transition, finding it recurrent for $d=2$ and transient for $d\geq 5$.
Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and
transience for $d\geq 4$. Additionally, we prove a 0-1 law for all $d$-ary
trees, and we exhibit a graph on which a 0-1 law does not hold.
  To prove recurrence when $d=2$, we construct a recursive distributional
equation for the number of visits to the root in a smaller process and show the
unique solution must be infinity a.s. The proof of transience when $d=5$ relies
on computer calculations for the transition probabilities of a large Markov
chain. We also include the proof for $d \geq 6$, which uses similar techniques
but does not require computer assistance.
