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.
