if the following are true:

• ((length of stack tree) - 1) - index = back_i
• the element at index back_i of stack tree = node (value, distance, previous)
• result of splitting pairs into nodes that exist in the tree tree and new nodes = pair (exists, new)
• result of updating nodes exists in tree tree with parent index index and parent distance distance = updated

then result of adding or updating children pairs of the node at backwards index index in tree tree = result of pushing nodes new into tree updated where parent is index and parent distance is distance