When vertices are unweighted and only edges have weights, treat edges as new vertices or assign edge weights to the deeper vertex.

Idea 1. Convert edges into new vertices. This is inefficient.

o--o--o  --> o-x-o-x-o

Idea 2. Assign edge weight to the deeper vertex. The is the internal implementation of fromEdges and LCAs are ignored on prod:

  o
  | <--- edge 1
  o <- write weight 1 here
  | <--- edge 2
  o <- write weight 2 here

>>> import AtCoder.Extra.Graph qualified as Gr
>>> import AtCoder.Extra.Tree.Hld qualified as Hld
>>> import AtCoder.Extra.Tree.TreeMonoid qualified as TM
>>> import Data.Semigroup (Sum (..))
>>> import Data.Vector.Unboxed qualified as VU
>>> -- 0--1--2--3
>>> --    +
>>> --    +--4--5
>>> let n = 6
>>> -- note that the edges must be bi-directed:
>>> let tree = Gr.build' n . Gr.swapDupe' $ VU.fromList [(0, 1), (1, 2), (2, 3), (1, 4), (4, 5)]
>>> let weights = VU.generate n Sum -- vertex `i` is given weight of `i`
>>> let hld = Hld.new tree
>>> tm <- TM.fromVerts hld {- `Sum` is commutative -} Commute weights
>>> TM.prod tm 1 3
Sum {getSum = 6}

>>> TM.prodSubtree tm 1
Sum {getSum = 15}

>>> TM.write tm 1 $ Sum 10
>>> TM.prod tm 1 3
Sum {getSum = 15}

>>> import AtCoder.Extra.Graph qualified as Gr
>>> import AtCoder.Extra.Tree.Hld qualified as Hld
>>> import AtCoder.Extra.Tree.TreeMonoid qualified as TM
>>> import Data.Semigroup (Sum (..))
>>> import Data.Vector.Unboxed qualified as VU
>>> -- 0--1--2--3
>>> --    +
>>> --    +--4--5
>>> let n = 6
>>> let edges = VU.fromList [(0, 1, Sum (1 :: Int)), (1, 2, Sum 2), (2, 3, Sum 3), (1, 4, Sum 4), (4, 5, Sum 5)]
>>> -- note that the edges must be bi-directed:
>>> let tree = Gr.build n $ Gr.swapDupe edges
>>> let hld = Hld.new tree
>>> let hld = Hld.new tree
>>> -- note that the edge doesn't have to be bi-directed:
>>> tm <- TM.fromEdges hld {- `Sum` is commutative -} Commute edges
>>> TM.prod tm 1 3
Sum {getSum = 5}

>>> TM.prodSubtree tm 1
Sum {getSum = 14}

>>> TM.write tm 2 $ Sum 10
>>> TM.prod tm 1 3
Sum {getSum = 13}