TITLE: Trees whose complement is also a tree
QUESTION [8 upvotes]: I am searching for those graphs $G$ where $G$ is a tree and its complement is also a tree. I came out with one such graph $P_4$. Are there any other too? I am not getting any other example. thanks

REPLY [4 votes]: Here is an alternative proof. If $n=1$ we are done, otherwise $n \geq 2$.
Any tree with at least two vertices has at least two leafs. Then the complement has at least two vertices of degree $n-2$.
Since the complement is also a tree, it has at least two vertices of degree $n-2$ and at least $2$ leafs. Then, the sum of those four degrees is
$$2(n-2)+2=2(n-1)$$
But by the Hanshaking Lemma the sum of all degrees is $2(n-1)$. Thus, (since the graph is connected) there cannot be any other vertex in the tree.
This shows that $n=4$, and the tree must have two vertices of degree $1$ and two vertices of degree $4-2=2$. It is easy to argue that this tree is $P_4$.