Easiest HLD on subtree

void dfs_sz(int u = 0)
{
    sz[u] = 1;
    for (auto &v : graph[u])
    {
        dfs_sz(v);
        sz[u] += sz[v];
        if (sz[v] > sz[ graph[u][0] ])
            swap(v, graph[u][0]);
    }
}

void dfs_hld(int u = 0)
{
    in[u] = t++;
    rin[ in[u] ] = u;
    for (auto v : graph[u])
    {
        nxt[v] = (v == g[u][0] ? nxt[u] : v);
        dfs_hld(v);
    }
    out[u] = t;
}

Then you will have such array that subtree of u corresponds to segment [in[u];out[u]) and the path from u to the last vertex in ascending heavy path from u (which is nxt[u]) will be [in[ nxt[u] ]; in[u]] subsegment which gives you the opportunity to process queries on paths and subtrees simultaneously in the same segment tree.
Sample Problem:
Codechef - Persistent oak