You are given a tree $T=(V,E)$ along with a designated root node $r \in V$.The parent of any node $v \ne r$, denoted $p(v)$, is defined to be the node adjacent to $v$ in the path from $r$ to $v$. By convention, $p(r) = r$.

For $k > 1$, define $p^k(v) = p^{k−1}(p(v))$ and $p^1(v) = p(v)$ (so $p^k(v)$ is the $k$-th ancestor of $v$).

Each vertex $v$ of the tree has an associated non-negative integer label $l(v)$. Give a linear-time algorithm to update the labels of all the vertices in T according to the following rule: $$l_{new}(v) = l(p^{l(v)}(v)).$$

This is problem 3.20 in Dasgupta's Algorithm. I think something needs to be done in the post-visit part of DFS traversal, but I don't know how to determine the $l(v)$-th parent of node $v$ for all $v$ in linear time.

Thanks very much.