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

There are ten labels. The successive deleted leaves of a graph are $3,4,5,1,6,7,2,9$, paired with the respective entries $1,1,1,2,2,2,8,8$. The final two labels are $8,10$. Thus the tree has edge setHere $ij$ denotes the unordered edge $\{i,j\}$; the last edge is written with a separator to distinguish label ten. Equivalently its adjacency lists are $1:\{2,3,4,5\}$, $2:\{1,6,7,8\}$, $8:\{2,9,10\}$, with all other vertices leaves of a graph. These lists make the constructed tree unambiguous.