On the uniform edge-partition of a tree

We study the problem of uniformly partitioning the edge set of a tree with n edges into k   connected components, where k⩽nk⩽n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k⩽4k⩽4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k  -split with ratio at most three in O(nlogk)O(nlogk) time. Experimental results on random trees are also shown.