We have a binary tree $T$ and for every node $v \in T$, we want to compute the new height if we select $v$ as the new root.

We consider the effect of making $v$ the new root. First, the height of the subtree of $v$ is still $l(v)$. Second, since $v$ is now the new root, it could go up to its parent and then to the farthest leaf. We define this distance as $r(v)$.