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

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.