 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/r/r2/iii/solution

A label appears in the code exactly its degree minus one times, as proved in (i). It is a leaf of a graph exactly when it never appears. Therefore

l = n - # {distinct labels appearing in the code} .

For example the code in (ii) uses three distinct labels, so its ten-vertex tree has seven leaves of a graph.

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