/**
 * @file heavy_ligt_decomposition.cpp
 * @brief Heavy-Light Decomposition implementation
 * @author: [Aniruthan R](https://github.com/aneee004)
 * @email: aniruth11052000@gmail.com
 * @CodeChef: aneee004
 * @CodeForces: aneee
 *
 * TODO: Support edge weight queries, by storing the edge weight value in it's child
 * algorithm verified by testing in CSES path queries: https://cses.fi/problemset/task/1138
*/

#include<iostream>
#include<vector>
#include<cstring>
#include<string>
#include<list>
#include<cmath>
#include<cassert>
#include<algorithm>
#include<numeric>

/**
 * Function Documentation: 
 * Heavy-Light Decomposition is a technique on trees, that supports the following:
 * 1: Update node s, with a value v
 * 2: Return the (sum) of all node values on the simple path from a to b
 * (sum) can also be replced with XOR, OR, AND, min, or max
 *
 * The update is done in O(log n) time, and
 * the query is done in O(log^2 n) time with HLD
 * where, n is the number of nodes
 *
 * The template type is the data type of the value stored in the nodes.
 * If a non-primitive data-type is used as a template, 
 * the coressponding operators must be overloaded.
 *
 * An HLD object can only be created with a constant number of nodes, and
 * it cannot be changed later. Creaty an empty instance is not supported.
 *
 * To start answering updates and queries,
 * 1. Create an instance of HLD<X> object (obj), with the required data type.
 * 2. Read in the edge/parent information and update it with obj.add_edge().
 * Note: The edges addes must be 0 indexed.
 * 3. Create a vector with initial node values, and call set_node_val() with it.
 * 4. Call obj.init() to populate the required information for supporting operations.
 * 5. Call obj.update(node, new_val), to update the value at index 'node' to the new value.
 * Note: node must be 0 indexed
 * 6. Call obj.query(a, b) to get the (sum) of node values in the simple path from a to b.
 * Note: a and b, must be 0 indexed.
 *
 * Sample I/O at the bottom.
 */

/**
 * A Basic Tree, which supports binary lifting
 * @tparam the data type of the values stored in the tree nodes
*/
template<typename X>
class Tree {
	// Deleting the default constructor
	// An instance can only be created with the number of nodes
	//
	// Defaults:
	// t_node indexing are zero based
	// t_root is 0
	// depth of root_node is 0
	//
	// Supports:
	// lift :- lift a node k units up the tree
	// kth_ancestor :- returns the kth ancestor
	// lca :- returns the least common ancestor
	
private:
	std::vector<std::list<int>> t_adj;
	const int t_nodes, t_maxlift;
	std::vector<std::vector<int>> t_par;
	std::vector<int> t_depth, t_size;
	int t_root;
	std::vector<X> t_val;
	template<typename T> friend class HLD;
	
	/**
	 * Utility function to compute sub-tree sizes
	 * @param u current dfs node
	 * @param p the parent of node @param u
	*/
	void dfs_size(int u, int p = -1) {
		for(const int& v : t_adj[u]) {
			if(v^p) {
				dfs_size(v, u);
				t_size[u] += t_size[v];
			}
		}
	}

	/**
	 * Utility function to populate the t_par vector
	 * @param u current dfs node
	 * @param p the parent of node @param u
	*/
	void dfs_lca(int u, int p = -1) {
		t_par[u][0] = p;
		if(p != -1) t_depth[u] = 1 + t_depth[p];
		for(int k = 1; k < t_maxlift; k++) {
			if(t_par[u][k-1] != -1) {
				t_par[u][k] = t_par[t_par[u][k-1]][k-1];
			}
		}

		for(const int& v: t_adj[u]) {
			if(v^p) {
				dfs_lca(v, u);
			}
		}
	}

public:
	/**
	 * Class parameterized constructor
	 * Resizes the and initilizes the data members
	 * @param nodes the total number of nodes in the tree
	*/
	explicit Tree(int nodes) 
		: t_nodes(nodes), t_maxlift(static_cast<int>(floor(log2(nodes))) + 1) {
		/* Initialize and resize all the vectors */
		t_root = 0; /* Default */
		t_adj.resize(t_nodes);
		t_par.assign(t_nodes, std::vector<int>(t_maxlift, -1));
		t_depth.assign(t_nodes, 0);
		t_size.assign(t_nodes, 1);
		t_val.resize(t_nodes);
	}
	
	/**
	 * Adds an undirected edge from node u to node v in the tree
	 * @param u the node where the edge is from
	 * @param v the node where the edge is to
	*/
	void add_edge(const int u, const int v) {
		t_adj[u].push_back(v);
		t_adj[v].push_back(u);
	}

	/**
	 * Set the root for the tree
	 * @param new_root the new root
	*/
	void change_root(int new_root) {
		t_root = new_root;
	}
	
	/**
	 * Set the values for all the nodes
	 * @param node_val a vector of size n, with all the node values
	 * where, n is the number of nodes
	*/
	void set_node_val(std::vector<X> node_val) {
		assert(static_cast<int>(node_val.size()) == t_nodes);
		t_val = node_val;
	}

	/**
	 * This function must be called after the tree adjacency list and node values are populated
	 * The function initializes the required parametes, and populates the segment tree
	*/
	void init() {
		assert(t_nodes > 0);
		dfs_size(t_root);
		dfs_lca(t_root);
	}

	/**
	 * The function lifts a node, k units up the tree
	 * @param p a pointer to the variable that stores the node id
	 * @param dist the distance to move up the tree
	 * @return void. The node if, after lifting is stored in the address pointed by @param p.
	*/
	void lift(int* const p, int dist) {
		for(int k = 0; k < t_maxlift; k++) {
			if(*p == -1) return;
			if(dist & 1) {
				*p = t_par[*p][k];
			}
			dist >>= 1;
		}
	} 
	
	/**
	 * The function returns the kth ancestor of a node
	 * @param p the node id whose kth ancestor is to be found
	 * @param dist the distance to move up the tree
	 * @return the kth ancestor of node @param p
	*/
	int kth_ancestor(int p, const int& dist) {
		lift(&p, dist);
		return p;
	}

	/**
	 * The function returns the least common ancestor of two nodes
	 * @param a node id_1
	 * @param b node id_2
	 * @return the least common ancestor of node @param a, and node @param b
	*/
	int lca(int a, int b) {
		assert(a >= 0 and b >= 0 and a < t_nodes and b < t_nodes);
		if(t_depth[a] > t_depth[b]) {
			lift(&a, t_depth[a] - t_depth[b]);
		}
		if(t_depth[b] > t_depth[a]) {
			lift(&b, t_depth[b] - t_depth[a]);
		}
		if(a == b) return a;
		for(int k = 0; k < t_maxlift; k++) {
			if(t_par[a][k] != t_par[b][k]) {
				a = t_par[a][k];
				b = t_par[b][k];
			}
		}
		return t_par[a][0];
	}
};

/** 
 * Segment Tree, to store heavy chains
 * @tparam the data type of the values stored in the tree nodes
*/
template<typename X>
class SG {
private:
	/**
	 * Everything here is private,
	 * and can only be acced thought the methods,
	 * in the derived class (HLD).
	*/

	std::vector<X> s_tree;
	int s_size;
	template<typename T> friend class HLD;
	X sret_init = 0;
	
	/**
	 * Function that specifies the type of operation involved when segments are combined
	 * @param lhs the left segment
	 * @param rhs the right segment
	 * @return the combined result
	*/
	X combine(X lhs, X rhs) {
		return lhs + rhs;
	}


	/**
	 * Class parameterized constructor. Resizes the and initilizes the data members.
	 * @param nodes the total number of nodes in the tree
	*/
	explicit SG(int size) {
		s_size = size;
		s_tree.assign(2*s_size, 0ll);
	}

	/**
	 * Update the value at a node
	 * @param p the node to be udpated
	 * @param v the update value
	*/
	void update(int p, X v) {
		for(p += s_size; p > 0; p >>= 1) s_tree[p] += v;
	}

	/**
	 * Make a range query from node label l to node label r
	 * @param l node label where the path starts
	 * @param r node label where the path ends
	*/
	X query(int l, int r) {
		X lhs = sret_init, rhs = sret_init;
		for(l += s_size, r += s_size + 1; l < r; l >>= 1, r >>= 1) {
			if(l & 1) lhs = combine(lhs, s_tree[l++]);
			if(r & 1) rhs = combine(s_tree[--r], rhs);
		}
		return combine(lhs, rhs);
	}
	
	/**
	 * Set the initialization for the query data type, based on requirement
	 * Change the sret_init, based on requirement:
	 * Sum Query: 0 (Default)
	 * XOR Query: 0 (Default)
	 * Min Query: Infinity 
	 * Max Query: -Infinity
	 * @param new_sret_init the new init
	*/
	void set_sret_init(X new_sret_init) {
		sret_init = new_sret_init;
	}
};

/** 
 * The Heavy-Light Decomposition class
 * @tparam the data type of the values stored in the tree nodes
*/
template<typename X>
class HLD : public Tree<X>, public SG<X> {
private:
	int label;
	std::vector<int> h_label, h_heavychlid, h_parent;

	/**
	 * Utility function to assign heavy child to each node (-1 for a leaf node)
	 * @param u current dfs node
	 * @param p the parent of node @param u
	*/
	void dfs_hc(int u, int p = -1) {
		int hc_size = -1, hc_id = -1;
		for(const int& v : Tree<X>::t_adj[u]) {
			if(v^p) {
				dfs_hc(v, u);
				if(Tree<X>::t_size[v] > hc_size) {
					hc_size = Tree<X>::t_size[v];
					hc_id = v;
				}
			}
		}
		h_heavychlid[u] = hc_id;
	}
	
	/**
	 * Utility function to assign highest parent that can be reached though heavy chains
	 * @param u current dfs node
	 * @param p the parent of node @param u
	*/
	void dfs_par(int u, int p = -1) {
		if(h_heavychlid[u] != -1) {
			h_parent[h_heavychlid[u]] = h_parent[u];
			dfs_par(h_heavychlid[u], u);
		}
		for(const int& v : Tree<X>::t_adj[u]) {
			if(v^p and v^h_heavychlid[u]) {
				dfs_par(v, u);
			}
		}
	}
	
	/**
	 * Utility function to lable the nodes so that heavy chains have a contigous lable
	 * @param u current dfs node
	 * @param p the parent of node @param u
	*/
	void dfs_labels(int u, int p = -1) {
		h_label[u] = label++;
		if(h_heavychlid[u] != -1) dfs_labels(h_heavychlid[u], u);
		for(const int& v : Tree<X>::t_adj[u]) {
			if(v^p and v^h_heavychlid[u]) {
				dfs_labels(v, u);
			}
		}
	}
	
	/**
	 * Utility function to break down a path query into two chain queries
	 * @param a node where the path starts
	 * @param b node where the path ends
	 * a and b must belong to a single root to leaf chain
	 * @return the sum of ndoe values in the simple path from a to b
	*/
	X chain_query(int a, int b) {
		X ret = SG<X>::sret_init;
		if(Tree<X>::t_depth[a] < Tree<X>::t_depth[b]) std::swap(a, b);
		while(Tree<X>::t_depth[a] >= Tree<X>::t_depth[b]) {
			int l = h_label[h_parent[a]];
			int r = h_label[a];
			if(Tree<X>::t_depth[h_parent[a]] < Tree<X>::t_depth[b]) {
				l += Tree<X>::t_depth[b] - Tree<X>::t_depth[h_parent[a]];
			}
			ret = SG<X>::combine(ret, SG<X>::query(l, r));
			a = Tree<X>::t_par[h_parent[a]][0];
			if(a == -1) break;
		}
		return ret;
	}
public:
	/**
	 * Class parameterized constructor. Resizes the and initilizes the data members.
	 * @param nodes the total number of nodes in the tree
	*/
	explicit HLD<X>(int nodes) : Tree<X>(nodes), SG<X>(nodes) { 
		/* Initialization and resize vectors */
		label = 0;
		h_label.assign(Tree<X>::t_nodes, -1);
		h_heavychlid.assign(Tree<X>::t_nodes, -1);
		h_parent.resize(Tree<X>::t_nodes);
		iota(h_parent.begin(), h_parent.end(), 0);
	}

	/**
	 * This function must be called after the tree adjacency list and node values are populated
	 * The function initializes the required parametes, and populates the segment tree
	*/
	void init() {
		Tree<X>::init();

		/**
		 * Fill the heavy child, greatest parent, and labels
		*/
		label = 0;
		dfs_hc(Tree<X>::t_root);
		dfs_par(Tree<X>::t_root);
		dfs_labels(Tree<X>::t_root);

		/**
		 * Segment Tree Initialization
		*/
		for(int i = 0; i < Tree<X>::t_nodes; i++) {
			SG<X>::s_tree[h_label[i] + Tree<X>::t_nodes] = Tree<X>::t_val[i];
		}
		for(int i = Tree<X>::t_nodes - 1; i > 0; i--) {
			SG<X>::s_tree[i] = SG<X>::combine(SG<X>::s_tree[i<<1], SG<X>::s_tree[i<<1|1]);
		}
	}

	/**
	 * This function updates the value at node with val
	 * @param node the node where the update is done
	 * @param val the value that is being updated
	*/
	void update(int node, X val) {
		X diff  = val - Tree<X>::t_val[node];
		SG<X>::update(h_label[node], diff);
		Tree<X>::t_val[node] = val;
	}

	/**
	 * This function returns the sum of node values in the simple path from from node_1 to node_2
	 * @param a the node where the simple path starts
	 * @param b the node where the simple path ends
	 * (parameters are interchangeable, i.e., the function is commutative)
	 * @return the sum of node values in the simple path from @param a to @param b
	*/
	X query(int a, int b) {
		int lc = Tree<X>::lca(a, b);
		X ret = SG<X>::sret_init;
		assert(lc != -1);
		ret += chain_query(a, lc);
		ret += chain_query(b, lc);
		return ret - Tree<X>::t_val[lc];
	}
};

/**
 * Test implementations
*/
void test_1() {
	std::cout << "Test 1:\n";
	/**
	 * Test details
	*/
	int n = 5;
	std::vector<int64_t> node_values = {4, 2, 5, 2, 1};
	std::vector<std::vector<int>> edges = {
		{1, 2},
		{1, 3},
		{3, 4},
		{3, 5}
	};
	std::vector<std::vector<int>> queries = {
		{2, 4},
		{1, 3, 2},
		{2, 4},
	};
	
	HLD<int64_t> hld(n);
	hld.set_node_val(node_values);
	for(int i = 0; i < n - 1; i++) {
		int u = edges[i][0], v = edges[i][1];
		hld.add_edge(u-1, v-1);
	}
	hld.init();
	for(const auto& q : queries) {
		int type = q[0];
		if(type == 1) {
			int p = q[1], x = q[2];
			hld.update(p - 1, x);
		}
		else if(type == 2) {
			int p = q[1];
			std::cout << hld.query(0, p - 1) << std::endl;
		}
		else {
			continue;
		}
	}
}

void test_2() {
	std::cout << "Test 2:\n";
	/**
	 * Test details (Bamboo)
	*/
	int n = 10;
	std::vector<int64_t> node_values = {1, 8, 6, 8, 6, 2, 9, 2, 3, 2};
	std::vector<std::vector<int>> edges = {
		{10, 5},
		{6, 2},
		{10, 7},
		{5, 2},
		{3, 9},
		{8, 3},
		{1, 4},
		{6, 4},
		{8, 7}
	};
	std::vector<std::vector<int>> queries = {
		{2, 10},
		{2, 6},
		{1, 3, 4},
		{2, 9},
		{1, 5, 3},
		{1, 7, 8},
		{2, 4},
		{2, 8},
		{1, 1, 4},
		{1, 2, 7}
	};
	
	HLD<int64_t> hld(n);
	hld.set_node_val(node_values);
	for(int i = 0; i < n - 1; i++) {
		int u = edges[i][0], v = edges[i][1];
		hld.add_edge(u-1, v-1);
	}
	hld.init();
	for(const auto& q : queries) {
		int type = q[0];
		if(type == 1) {
			int p = q[1], x = q[2];
			hld.update(p - 1, x);
		}
		else if(type == 2) {
			int p = q[1];
			std::cout << hld.query(0, p - 1) << std::endl;
		}
		else {
			continue;
		}
	}
}

/**
 * Main function
*/
int main() {
	test_1();
	test_2();
	return 0;
}

/*
 * Sample Output 1:
 * 11
 * 8
 *
 * Sample Output 2:
 * 27
 * 11
 * 45
 * 9
 * 34
*/