if the following are true:

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

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