3558 - Number of Ways to Assign Edge Weights I.

Medium

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.

Select any one node x at the maximum depth. Return the number of ways to assign edge weights in the path from node 1 to x such that its total cost is odd.

Since the answer may be large, return it modulo <code>10⁹ + 7</code>.

Note: Ignore all edges not in the path from node 1 to x.

Example 1:

Input: edges = [1,2]

Output: 1

Explanation:

• 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]

Output: 2

Explanation:

• The maximum depth is 2, with nodes 4 and 5 at the same depth. Either node can be selected for processing.

• For example, 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.

Constraints:

• <code>2 <= n <= 10⁵</code>

• edges.length == n - 1

• <code>edgesi == u_i, v_i</code>

• <code>1 <= u_i, v_i<= n</code>

• edges represents a valid tree.