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3558. Number of Ways to Assign Edge Weights I

Description

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 edges[i] = [ui, vi] indicates that there is an edge between nodes ui and vi.

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 109 + 7.

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:

  • 2 <= n <= 105
  • edges.length == n - 1
  • edges[i] == [ui, vi]
  • 1 <= ui, vi <= n
  • edges represents a valid tree.

Solutions

Solution 1

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