In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=\lceil\frac{n}{2}\rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M = \{m_1,m_2,...,m_x\}$ where $m_i\le(n-1)$ and $\Sigma_{i=1}^{x} m_i$ = $\frac{n(n-1)}{2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.

翻译：本研究探讨了将无向、无权完全图（$K_n$，阶数为$n$，边数为$m$）方法性地分解为最少数量边不相交树的问题。我们发现，当将$K_n$的边集分解为规模序列$M = \{m_1,m_2,...,m_x\}$的边不相交树（其中$m_i\le(n-1)$且$\Sigma_{i=1}^{x} m_i$ = $\frac{n(n-1)}{2}$）时，正整数$x$取最小值且$x=\lceil\frac{n}{2}\rceil$。为将$K_n$的边集分解为最少数量的边不相交树，我们提出的算法总时间复杂度为$O(m)$。