3559 - Number of Ways to Assign Edge Weights II.

Hard

There is an undirected tree with n nodes labeled from 1 to n, rooted at node 1. The tree is represented by a 2D integer array edges of length n - 1, where <code>edgesi = u_i, v_i</code> indicates that there is an edge between nodes <code>u_i</code> and <code>v_i</code>.

Initially, all edges have a weight of 0. You must assign each edge a weight of either 1 or 2.

The cost of a path between any two nodes u and v is the total weight of all edges in the path connecting them.

You are given a 2D integer array queries. For each <code>queriesi = u_i, v_i</code>, determine the number of ways to assign weights to edges in the path such that the cost of the path between <code>u_i</code> and <code>v_i</code> is odd.

Return an array answer, where answer[i] is the number of valid assignments for queries[i].

Since the answer may be large, apply modulo <code>10⁹ + 7</code> to each answer[i].

Note: For each query, disregard all edges not in the path between node <code>u_i</code> and <code>v_i</code>.

Example 1:

Input: edges = [1,2], queries = [1,1,1,2]

Output: 0,1

Explanation:

• Query [1,1]: The path from Node 1 to itself consists of no edges, so the cost is 0. Thus, the number of valid assignments is 0.

• Query [1,2]: The path from Node 1 to Node 2 consists of one edge (1 → 2). Assigning weight 1 makes the cost odd, while 2 makes it even. Thus, the number of valid assignments is 1.

Example 2:

Input: edges = [1,2,1,3,3,4,3,5], queries = [1,4,3,4,2,5]

Output: 2,1,4

Explanation:

• Query [1,4]: The path from Node 1 to Node 4 consists of two edges (1 → 3 and 3 → 4). Assigning weights (1,2) or (2,1) results in an odd cost. Thus, the number of valid assignments is 2.

• Query [3,4]: The path from Node 3 to Node 4 consists of one edge (3 → 4). Assigning weight 1 makes the cost odd, while 2 makes it even. Thus, the number of valid assignments is 1.

• Query [2,5]: The path from Node 2 to Node 5 consists of three edges (2 → 1, 1 → 3, and 3 → 5). Assigning (1,2,2), (2,1,2), (2,2,1), or (1,1,1) makes the cost odd. Thus, the number of valid assignments is 4.

Constraints:

• <code>2 <= n <= 10⁵</code>

• edges.length == n - 1

• <code>edgesi == u_i, v_i</code>

• <code>1 <= queries.length <= 10⁵</code>

• <code>queriesi == u_i, v_i</code>

• <code>1 <= u_i, v_i<= n</code>

• edges represents a valid tree.