Link cut trees. They can be used to query information about paths in a forest, while also allowing you to add and remove edges in the forest. For example, you

Link cut trees. They can be used to query information about paths in a forest, while also allowing you to add and remove edges in the forest. For example, you could use them to store the minimum spanning tree of a graph, while supporting updates if edges are added to the graph. They can also be used to speed up max flow algorithms by finding the augmenting path in logarithmic time.