In the 1997 movie Good Will Hunting, Professor Lambeau challenges his students with a complex combinatorial problem involving series-reduced trees. You can imagine how shocked he is when he catches the prodigious self-taught mathematician and janitor, Will Hunting solving the problem on a blackboard.
The problem presented in the movie is to find the number of series-reduced trees (referred to as “homeomorphically irreducible trees” in the firm) with 10 vertices. A tree can be defined as a connected graph with no cycles.
A series-reduced tree is a tree that does not contain any vertices of degree 2. In other words, every vertex in the tree must either be a leaf (degree 1) or have a degree of at least 3. We can ignore superficial changes in rotations or angles of branches.
Although Will only draws 8 of them, there are 10 solutions to this problem:
