if the following are true:

• children of the node at backwards index i of tree tree in graph graph = children
• result of adding or updating children children of the node at backwards index i in tree tree = new_tree

then output of the shortest_path function where the input graph is graph, backwards index is i, and tree is tree = output of the shortest_path function where the input graph is graph, backwards index is (i + 1), and tree is new_tree