Sparse Table Idea

Define up[v][j] = the $2^j$-th ancestor of node $v$. This is the node you reach by jumping up $2^j$ steps from $v$. For example, up[v][0] is the parent (jump $2^0 = 1$ step). up[v][1] is the grandparent (jump $2^1 = 2$ steps). up[v][3] is the $8$th ancestor.

If you precompute this table, you can answer any $k$-th ancestor query by decomposing $k$ into powers of $2$ and making at most $\log k$ jumps.