Chain Positions

Within each chain, assign positions from top to bottom. Let pos[v] be the position of node $v$ in its chain, starting from $0$. The topmost node in a chain has $pos = 0$. As you DFS, maintain a counter chain_pos[c] for each chain $c$. When you visit node $v$ in chain $c$, set pos[v] = chain_pos[c] and increment chain_pos[c].

Now each node has two coordinates: which chain it is in via chain[v], and where in that chain via pos[v]. This lets you map tree nodes to segment tree indices.