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18 April, 17:14

How many trees are there whose complement is also a tree?

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  1. 18 April, 17:56
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    There is only one tree whose complement is also a tree. Using the graph theory and discrete mathematics we can prove this.

    First, consider that for a tree that has n vertices, the number of edges is (n - 1)

    Next, for n vertices, there are n (n - 1) / 2 unordered pairs.

    To find the complement of the tree, this equation must hold true:

    n (n - 1) / 2 - (n - 1) = (n - 1)

    Simplifying

    n (n - 1) - 2 (n - 1) = 2 (n - 1)

    n² - n - 2n + 2 - 2n + 2 = 0

    n² - 5n + 4 = 0

    Solving the quadratic equation

    n = 1 and 4

    A tree can't be on a vertex of only 1. So the only applicable number of vertices is 4. So, there can only be one true whose complement is a tree.
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