Core Idea - Same Depth First

If $a$ and $b$ have different depths, bring the deeper one up to the depth of the shallower one. Compute depth[v] for all nodes (via BFS or DFS).

If depth[a] > depth[b], jump $a$ up by depth[a] - depth[b] steps. Now $a$ and $b$ are at the same depth. Use binary lifting to perform the jump in $O(\log n)$ time. After this step, $a$ and $b$ are at the same level and you can begin searching for their meeting point.