The decomposition walk

Second DFS, preorder, heavy child first:

pos[u]  = cnt++;                          // entry order
head[u] = (hv[par[u]] == u) ? head[par[u]] : u;

Heavy-first entry order makes every chain a contiguous range of positions — a chain [1,2,4] gets pos 0,1,2 on the hand tree. The segment tree is built over POSITIONS: a[pos[v]] = val[v].

The path query loop:

while (head[u] != head[v]) {                if (dep[head[u]] < dep[head[v]]) swap(u, v);
    apply(pos[head[u]], pos[u]);          // whole chain head..u
    u = par[head[u]];                     // jump above the chain
}
if (pos[u] > pos[v]) swap(u, v);
apply(pos[u], pos[v]);                    // final same-chain segment

Lift the DEEPER HEAD (not the deeper node). Two classic real bugs: forgetting the final same-chain segment (entirely, or its swap), and indexing the tree by vertex id instead of pos. Both pass the tiny sample, both die on any real test.