We present algorithms for computing hierarchical decompositions of trees satisfying different optimization criteria, including balanced cluster size, bounded number of clusters, and logarithmic depth of the decomposition. Furthermore, every high-level representation of the tree obtained from such decompositions is guaranteed to be a tree. These criteria are relevant in many application settings, but appear to be difficult to achieve simultaneously. Our algorithms work by vertex deletion and hinge upon the new concept of t-divider, that generalizes the well-known concepts of centroid and separator. The use of t-dividers, combined with a reduction to a classical scheduling problem, yields an algorithm that, given a n-vertex tree T, builds in O(nlogn) worst-case time a hierarchical decomposition of T satisfying all the aforementioned requirements. © 2003 Published by Elsevier B.V.
Finocchi, I., & Petreschi, R. (2004). Divider-based algorithms for hierarchical tree partitioning. In Discrete Applied Mathematics (Vol. 136, pp. 227–247). https://doi.org/10.1016/S0166-218X(03)00443-8