# Heavy Light 2 White Falcon

### Problem Statement :

```White Falcon was amazed by what she can do with heavy-light decomposition on trees. As a resut, she wants to improve her expertise on heavy-light decomposition. Her teacher gave her an another assignment which requires path updates. As always, White Falcon needs your help with the assignment.

You are given a tree with N  nodes and each node's value Vi is initially 0.

Let's denote the path from node u to node v  like this: p1, p2, p3 . . . pk, where p1 = u  and pk = v, and pi and   pi+1 are connected.

The problem asks you to operate the following two types of queries on the tree:

"1 u v x" Add  x to  valp1, 2x to  valp2, 3x to valp3 , ...,  kx to valpk.
"2 u v" print the sum of the nodes' values on the path between u and v  at modulo 10^9 + 7 .

Input Format

First line cosists of two integers N and Q seperated by a space.
Following N - 1  lines contains two integers which denote the undirectional edges of the tree.
Following   Q lines contains one of the query types described above.

Note: Nodes are numbered by using 0-based indexing.

Constraints

1  <=  N , Q  <=  50000
0  <=  x  <= 10^9 + 7

Output Format

For every query of second type print a single integer.```

### Solution :

```                            ```Solution in C :

In   C++  :

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <cassert>

using namespace std;

const long long mod = 1000000007ll;
const long long inv2 = 500000004ll;
const int N = 50010;

long long calc(long long _n, long long _a, long long _d) {
long long ans = (_a + _a + (_n-1) * _d) % mod;
ans = (ans * _n) % mod;
ans = (ans * inv2) % mod;
return ans;
}

class node {
public:
int l, r;
long long sum, a, d;
node *left, *right;

void propagate(void) {
if(a == 0 and d == 0) return;
int mid = (l + r) / 2;
left->update(l, mid, a, d);
long long nw_a = (a + (mid - l + 1) * d) % mod;
right->update(mid+1, r, nw_a, d);
sum = (left->sum + right->sum) % mod;
a = d = 0ll;
}

void update(int x, int y, long long _a, long long _d) {
if(y < l or r < x) return;
if(x <= l and r <= y) {
sum = (sum + calc(r - l + 1, _a, _d)) % mod;
a = (a + _a) % mod;
d = (d + _d) % mod;
return;
}
propagate();
int mid = (l + r) / 2;
if(y <= mid) {
left->update(x, y, _a, _d);
}else if(mid < x) {
right->update(x, y, _a, _d);
}else {
left->update(x, mid, _a, _d);
long long nw_a = (_a + (mid - x + 1) * _d) % mod;
right->update(mid+1, y, nw_a, _d);
}
sum = (left->sum + right->sum) % mod;
}

long long query(int x, int y) {
if(y < l or r < x) return 0ll;
if(x <= l and r <= y) return sum;
propagate();
return (left->query(x, y) + right->query(x, y)) % mod;
}

node(int _l, int _r) : l(_l), r(_r), sum(0ll), a(0ll), d(0ll) {}
};

node* init(int l, int r) {
node *p = new node(l, r);
if(l < r) {
int mid = (l + r) / 2;
p->left = init(l, mid);
p->right = init(mid+1, r);
}
return p;
}

int n, q;

vector<int> Path[N];
int G[N], H[N], P[N], pos[N], sz[N];

void dfs_init(int u, int p, int h) {
H[u] = h;
P[u] = p;
sz[u] = 1;
if(v == p) continue;
dfs_init(v, u, h+1);
sz[u] += sz[v];
}
}
void dfs_HLD(int u) {
Path[u].push_back(u);
for(int i = 0;i < Path[u].size();i++) {
int v = Path[u][i];
G[v] = u;
pos[v] = i;
if(vv == P[v]) continue;
if(2*sz[vv] >= sz[v]) {
Path[u].push_back(vv);
}else {
dfs_HLD(vv);
}
}
}
head[G[u]] = init(0, Path[u].size() - 1);
}
int lca(int u, int v) {
while(G[u] != G[v]) {
if(H[G[u]] < H[G[v]]) swap(u, v);
u = P[G[u]];
}
return pos[u] < pos[v] ? u : v;
}
void update(int u, int v, long long a, long long d) {
int l = lca(u, v);
while(G[u] != G[l]) {
a = (a + (pos[u] + 1) * d) % mod;
head[G[u]]->update(0, pos[u], (a-d+mod) % mod, mod-d);
u = P[G[u]];
}
if(pos[l] + 1 <= pos[u]) {
a = (a + (pos[u] - pos[l]) * d) % mod;
head[G[u]]->update(pos[l]+1, pos[u], (a-d+mod) % mod, mod-d);
}
long long nw_a = (a + (H[v] - H[l]) * d) % mod, nw_d = mod - d;
while(G[v] != G[l]) {
nw_a = (nw_a + (pos[v] + 1) * nw_d) % mod;
head[G[v]]->update(0, pos[v], (nw_a + d) % mod, d);
v = P[G[v]];
}
nw_a = (nw_a + (pos[v] - pos[l]) * nw_d) % mod;
assert(a == nw_a);
}
long long query(int u, int v) {
long long ans = 0ll;
while(G[u] != G[v]) {
if(H[G[u]] < H[G[v]]) {
swap(u, v);
}
ans = (ans + head[G[u]]->query(0, pos[u])) % mod;
u = P[G[u]];
}
if(pos[u] > pos[v]) swap(u, v);
ans = (ans + head[G[u]]->query(pos[u], pos[v])) % mod;
ans = (ans + mod) % mod;
return ans;
}
int main() {

ios::sync_with_stdio(false);

cin >> n >> q;
for(int i = 0;i < n-1;i++) {
int u, v;
cin >> u >> v;
}

dfs_init(0, 0, 0);
dfs_HLD(0);

for(int i = 0;i < q;i++) {
int type;
cin >> type;
if(type == 1) {
int u, v;
long long x;
cin >> u >> v >> x;
update(u, v, x, x);
}else {
int u, v;
cin >> u >> v;
cout << query(u, v) << "\n";
}
}

return 0;
}

In   Java  :

import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.List;
import java.util.Queue;
import java.util.Scanner;

public class Solution {
static List<Integer>[] conn;
static long[] v;
static long[] ov;
static int[] d;
static int[] p;
static int count;

public static void main(String[] args) {

Scanner sc = new Scanner(System.in);
int n = sc.nextInt();
int q = sc.nextInt();
int r = 0;
long time = System.currentTimeMillis();
count = n;
conn = new List[n + 1];
v = new long[n + 1];
ov = new long[n + 1];
d = new int[n + 1];
p = new int[n + 1];

for (int i = 0; i < n - 1; i++) {
int x = sc.nextInt();
int y = sc.nextInt();

List<Integer> xconn = conn[x];
if (xconn == null) {
xconn = new ArrayList<Integer>();
conn[x] = xconn;
}

List<Integer> yconn = conn[y];
if (yconn == null) {
yconn = new ArrayList<Integer>();
conn[y] = yconn;
}

}

d[r] = 1;
treefy3(r);

for (int i = 0; i < q; i++) {
String uq = sc.next();

if ("1".equals(uq)) {
int u = sc.nextInt();
int v = sc.nextInt();
int x = sc.nextInt();

update2(u, v, x);
} else {
int u = sc.nextInt();
int v = sc.nextInt();

long s = sum(u, v);
System.out.println(s % 1000000007);
}
}

//        System.out.println(System.currentTimeMillis() - time);

}

private static void update(int un, int vn, long x) {
// List<Integer> vl = new ArrayList<>(count);

int[] va = new int[count];
int idx = 0;

int ud = d[un];
int vd = d[vn];

int rn = un;
boolean isUnUsed = true;
if (ud <= vd) {
rn = vn;
isUnUsed = false;
}

int k = 1;
int abs = Math.abs(vd - ud);
for (int i = 0; i < abs; i++) {
if (isUnUsed) {
// v[rn] += k++ * x;
v[rn] = (v[rn] + k++ * x) % 1000000007;
} else {
va[idx++] = rn;

ov[rn] = v[rn];
v[rn] = (v[rn] + (ud + d[rn] - 1) * x) % 1000000007;
}
rn = p[rn];
}

if (d[un] <= d[vn]) {
vn = rn;
} else {
un = rn;
}

while (un != vn) {
// v[un] += k++ * x;
v[un] = (v[un] + k++ * x) % 1000000007;
va[idx++] = vn;
ov[vn] = v[vn];
v[vn] = (v[vn] + (ud + d[vn] - 1) * x) % 1000000007;
un = p[un];
vn = p[vn];
}

// v[un] += k++ * x;
v[un] = (v[un] + k++ * x) % 1000000007;

int vs = idx;

if (d[un] != 1) {
for (int i = vs - 1; i >= 0; i--) {
int n = va[i];
// n.value += k++ * x;
v[n] = (ov[n] + k++ * x) % 1000000007;
}
}
}

private static void update2(int un, int vn, long x) {
int ud = d[un];
int vd = d[vn];

int uns = un;
int vns = vn;

int k = 1;
if (ud > vd) {
int abs = ud - vd;
for (int i = 0; i < abs; i++) {
v[uns] = (v[uns] + k++ * x) % 1000000007;
uns = p[uns];
}
}else{
int abs = vd - ud;
for (int i = 0; i < abs; i++) {
vns = p[vns];
}
}

while (uns != vns) {
// v[un] += k++ * x;
v[uns] = (v[uns] + k++ * x) % 1000000007;
uns = p[uns];
vns = p[vns];
}

// v[un] += k++ * x;
v[uns] = (v[uns] + k * x) % 1000000007;

int tt = k + (d[vn] - d[vns]);

while(vns != vn) {

// n.value += k++ * x;
v[vn] = (v[vn] + tt-- * x) % 1000000007;
vn = p[vn];
}
}

private static long sum(int un, int vn) {
long sum = 0;

int rn = un;
if (d[un] <= d[vn]) {
rn = vn;
}

int abs = Math.abs(d[vn] - d[un]);
for (int i = 0; i < abs; i++) {
sum += v[rn];
rn = p[rn];
}

if (d[un] <= d[vn]) {
vn = rn;
} else {
un = rn;
}

while (un != vn) {
sum += v[un] + v[vn];
un = p[un];
vn = p[vn];
}
sum += v[un];

return sum;
}

static int c = 1;

private static void treefy2(int rn) {
Queue<Integer> q = new ArrayDeque<>();

while (!q.isEmpty()) {
int n = q.poll();

int s = conn[n].size();
int dd = d[n] + 1;
for (int i = 0; i < s; i++) {
int cn = conn[n].get(i);

if (d[cn] == 0) {
p[cn] = n;
d[cn] = dd;
}
}
}
}

private static void treefy3(int rn) {
int[] iq = new int[count];
int idx = 0;
iq[idx] = rn;

while (idx >= 0) {
int n = iq[idx];
idx--;
int s = conn[n].size();
int dd = d[n] + 1;
for (int i = 0; i < s; i++) {
int cn = conn[n].get(i);

if (d[cn] == 0) {
p[cn] = n;
d[cn] = dd;
idx++;
iq[idx] = cn;
}
}
}
}
}

In  Python3 :

from operator import attrgetter

MOD = 10**9 + 7

def solve(edges, queries):
nodes, leaves = make_tree(edges)
hld(leaves)

results = []
for query in queries:
if query[0] == 1:
update(nodes[query[1]], nodes[query[2]], query[3])
elif query[0] == 2:
results.append(sum_range(nodes[query[1]], nodes[query[2]]))

return results

def make_tree(edges):
nodes = [
Node(i)
for i in range(len(edges) + 1)
]

# the tree is a graph for now
# as we don't know the direction of the edges
for edge in edges:
nodes[edge[0]].children.append(nodes[edge[1]])
nodes[edge[1]].children.append(nodes[edge[0]])

# pick the root of the tree
root = nodes[0]
root.depth = 0

# for each node, remove its parent of its children
stack = []
leaves = []
for child in root.children:
stack.append((child, root, 1))
for node, parent, depth in stack:
node.children.remove(parent)
node.parent = parent
node.depth = depth

if len(node.children) == 0:
leaves.append(node)
continue

for child in node.children:
stack.append((child, node, depth + 1))

return nodes, leaves

def hld(leaves):
leaves = sorted(leaves, key=attrgetter('depth'), reverse=True)

for leaf in leaves:
leaf.chain = Chain()
leaf.chain_i = 0

curr_node = leaf
while curr_node.parent is not None:
curr_chain = curr_node.chain
if curr_node.parent.chain is not None:
curr_chain.init_fenwick_tree()
curr_chain.parent = curr_node.parent.chain
curr_chain.parent_i = curr_node.parent.chain_i
break

curr_node.parent.chain = curr_chain
curr_node.parent.chain_i = curr_chain.size
curr_node.chain.size += 1
curr_node = curr_node.parent

if curr_node.parent is None:
curr_chain.init_fenwick_tree()

def update(node1, node2, x):
path_len = 0
chain1 = node1.chain
chain_i1 = node1.chain_i
depth1 = node1.depth
chains1 = []
chain2 = node2.chain
chain_i2 = node2.chain_i
depth2 = node2.depth
chains2 = []

while chain1 is not chain2:
step1 = chain1.size - chain_i1
step2 = chain2.size - chain_i2

if depth1 - step1 > depth2 - step2:
path_len += step1
chains1.append((chain1, chain_i1))
depth1 -= step1
chain_i1 = chain1.parent_i
chain1 = chain1.parent
else:
path_len += step2
chains2.append((chain2, chain_i2))
depth2 -= step2
chain_i2 = chain2.parent_i
chain2 = chain2.parent

path_len += abs(chain_i1 - chain_i2) + 1

curr_val1 = 0
for (chain, chain_i) in chains1:
curr_val1 += (chain.size - chain_i) * x

curr_val2 = (path_len + 1) * x
for (chain, chain_i) in chains2:
curr_val2 -= (chain.size - chain_i) * x

if chain_i1 <= chain_i2:
else:

def sum_range(node1, node2):
sum_ = 0
chain1 = node1.chain
chain_i1 = node1.chain_i
depth1 = node1.depth
chain2 = node2.chain
chain_i2 = node2.chain_i
depth2 = node2.depth
while chain1 is not chain2:
step1 = chain1.size - chain_i1
step2 = chain2.size - chain_i2
if depth1 - step1 > depth2 - step2:
sum_ += chain1.ftree.range_sum(chain_i1, chain1.size - 1)

depth1 -= step1
chain_i1 = chain1.parent_i
chain1 = chain1.parent
else:
sum_ += chain2.ftree.range_sum(chain_i2, chain2.size - 1)

depth2 -= step2
chain_i2 = chain2.parent_i
chain2 = chain2.parent

if chain_i1 > chain_i2:
chain_i1, chain_i2 = chain_i2, chain_i1

sum_ += chain1.ftree.range_sum(chain_i1, chain_i2)

return int(sum_ % MOD)

class Node():
__slots__ = ['i', 'val', 'parent', 'children', 'depth', 'chain', 'chain_i']

def __init__(self, i):
self.i = i
self.val = 0
self.parent = None
self.depth = None
self.children = []
self.chain = None
self.chain_i = -1

class Chain():
__slots__ = ['size', 'ftree', 'parent', 'parent_i']

def __init__(self):
self.size = 1
self.ftree = None
self.parent = None
self.parent_i = -1

def init_fenwick_tree(self):
self.ftree = RURQFenwickTree(self.size)

def g(i):
return i & (i + 1)

def h(i):
return i | (i + 1)

class RURQFenwickTree():
def __init__(self, size):
self.tree1 = RUPQFenwickTree(size)
self.tree2 = RUPQFenwickTree(size)
self.tree3 = RUPQFenwickTree(size)

def add(self, l, r, k, x):
k2 = k * 2
self.tree2.add(l, (3 - 2*l) * x + k2)
self.tree2.add(r+1, -((3 - 2*l) * x + k2))
self.tree3.add(l, (l**2 - 3*l + 2) * x + k2 * (1 - l))
self.tree3.add(r+1, (r**2 + 3*r - 2*r*l) * x + k2 * r)

def prefix_sum(self, i):
sum_ = i**2 * self.tree1.point_query(i)
sum_ += i * self.tree2.point_query(i)
sum_ += self.tree3.point_query(i)

return ((sum_ % (2 * MOD)) / 2) % MOD

def range_sum(self, l, r):
return self.prefix_sum(r) - self.prefix_sum(l - 1)

class RUPQFenwickTree():
def __init__(self, size):
self.size = size
self.tree = [0] * size

j = i
while j < self.size:
self.tree[j] += x
j = h(j)

def point_query(self, i):
res = 0
j = i
while j >= 0:
res += self.tree[j]
j = g(j) - 1

return res

if __name__ == '__main__':
nq = input().split()

n = int(nq[0])

q = int(nq[1])

tree = []

for _ in range(n-1):
tree.append(list(map(int, input().rstrip().split())))

queries = []

for _ in range(q):
queries.append(list(map(int, input().rstrip().split())))

results = solve(tree, queries)

print('\n'.join(map(str, results)))```
```

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