Let $G=(V,E)$ be an edge-weighted graph, and let
$w(H)$ denote the sum of the weights of the edges in a subgraph
$H$ of $G$. Given a positive integer $k$, the balanced tree
partitioning problem requires to cover all vertices in $V$ by a set
$\mathcal{T}$ of $k$ trees of the graph so that the
ratio $\alpha$ of $\max_{T\in \mathcal{T}}w(T)$ to
$w(T^*)/k$ is minimized, where $T^*$ denotes
a minimum spanning tree of $G$.
The problem has been used as a core analysis in designing
approximation algorithms for several types of graph partitioning
problems over metric spaces, and the performance guarantees depend
on the ratio $\alpha$ of the corresponding balanced tree
partitioning problems. It is known that the best possible value of
$\alpha$ is 2 for the general metric space.
In this paper, we study the problem in the $d$-dimensional
Euclidean space $\mathbb{R}^d$,
and break the bound 2 on $\alpha$, showing that $\alpha
<2\sqrt{3}-3/2 \fallingdotseq 1.964$ for $d\geq
3$ and $\alpha <(13 + \sqrt{109})/12 \fallingdotseq 1.953$ for $d=2$.
These new results enable us to directly
improve the performance guarantees of several existing
approximation algorithms for
graph partitioning problems if the metric space is an Euclidean space.