  We present a very simple bijective proof of Cayley's formula due to Foata and
Fuchs (1970). This bijection turns out to be very useful when seen through a
probabilistic lens; we explain some of the ways in which it can be used to
derive probabilistic identities, bounds, and growth procedures for random trees
with given degrees, including random d-ary trees. We also introduce a partial
order on the degree sequences of rooted trees, and conjecture that it induces a
stochastic partial order on heights of random rooted trees with given degrees.
