Suppose I have a binary tree $G = (V, E)$ (with undirected edges) that is partitioned into sets of k vertices, where each set of vertices is a connected subgraph of $G$. Additionally, if there are sets that have less than k vertices, then at least one of these vertices must be a leaf.
Now suppose I wanted to increase the size of each set to be $k+1$ and hold all the above properties. I'm not entirely sure what tools I would use to construct such an algorithm. Any ideas or resources you can link me to?