Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / R / R2 / i / Solution

Encode a labelled tree by repeatedly deleting its smallest-labelled leaf of a graph and recording its neighbor, stopping when two vertices remain. This produces a sequence of length $n-2$ with entries in $\{1,\dots ,n\}$.

Conversely, given such a sequence, at each step choose the smallest unused label not occurring in the remaining sequence, join it to the first sequence entry, delete that chosen label and the first entry, and continue. Finally join the two labels left over. A label outside the remaining sequence always exists because two more labels than entries remain. The process never uses a previously deleted label as a sequence entry. Each step attaches a new leaf of a graph to the eventual remaining graph, so backward induction shows that the result is connected and acyclic, hence a tree.

In a tree encoding, a vertex occurs once for every incident edge except the edge by which it is eventually removed, or the final surviving edge. Thus its number of occurrences is its degree minus one. The labels absent from the current remaining sequence are exactly the current leaves of a graph. Both procedures therefore choose the same smallest leaf of a graph and edge at every stage. They are inverse maps, proving the claimed bijection, including the empty sequence when $n=2$.