We introduce a graph partitioning problem motivated by computational topology and propose two algorithms that produce approximate solutions. Specifically, given a weighted, undirected graph $G$ and a positive integer $k$, we desire to find $k$ disjoint trees within $G$ such that each vertex of $G$ is contained in one of the trees and the weight of the largest tree is as small as possible. We are unable to find this problem in the graph partitioning literature, but we show that the problem is NP-complete. We then propose two approximation algorithms, one that uses a spectral clustering approach and another that employs a dynamic programming strategy, which produce near-optimal partitions on a family of test graphs. We describe these algorithms and analyze their empirical performance.