How can I find all equal paths in a degenerate tree from a certain starting vertex?

I have degenerate tree

(it looks like an array or a double linked list). For example, this tree:

Each edge has some weight. I want to find all equal paths that start at each vertex.

Let the edges have the following weights (this is just an example):

a-b = 3

b-c = 1

c-d = 1

d-e = 1

Then:

• The vertex A
  
  has no equal path (no vertex on the left side).
• The vertex B
  
  has one equal pair. The path B-A
  
  is equal to the path B-E

(3 == 3)
  
  .
• The vertex C
  
  has one equal pair. The path B-C
  
  is equal to the path C-D

(1 == 1)
  
  .
• The vertex D
  
  has one equal pair. The path C-D
  
  is equal to the path D-E

(1 == 1)
  
  .
• The vertex E
  
  has no equal path (no vertex on the right side).

I am implementing a simple algorithm that works in O(n^2)

. But it's too slow for me.