(i) We are given a finite set of airports. Assume that between any two airports, $i$ and $j$, there are $a_{i j}=a_{j i}$ flights in each direction on every day. A confused traveller takes one flight per day, choosing at random from all available flights. Starting from $i$, how many days on average will pass until the traveller returns again to $i$ ? Be careful to allow for the case where there may be no flights at all between two given airports.

(ii) Consider the infinite tree $T$ with root $R$, where, for all $m \geqslant 0$, all vertices at distance $2^{m}$ from $R$ have degree 3 , and where all other vertices (except $R$ ) have degree 2 . Show that the random walk on $T$ is recurrent.