# Is a subtree of a minimum spanning tree a minimum spanning tree of the subgraph spanned by the subtree?

Let $$G$$ be a connected weighted undirected graph. Let $$T$$ be a minimum spanning tree (MST) of $$G$$. Consider removing an edge $$e=(a,b)$$ from $$T$$, which will give two subtrees $$T_a$$ and $$T_b$$, where $$Ta$$ contains the vertex $$a$$ and $$T_b$$ contains the vertex $$b$$. Now consider the subgraph of $$G$$, $$G_a$$, which contains the vertices of $$G$$ that are in $$T_a$$, and the edges of $$G$$ that have both endpoints in $$T_a$$

Is $$T_a$$ an MST for the subgraph $$G_a$$?

Intuitively, I believe $$T_a$$ is an MST for the subgraph $$G_a$$, but I'm having a lot of difficulty proving the result.

Let’s prove this by contradiction. Say $$T_a$$ is not an MST of $$G_a$$, that means there is some cheaper MST $$T_a^*$$. However, if that were true, then by reconnecting $$T_a^*$$ to $$T_b$$ we would obtain a tree that spans $$G_a$$ that has a lower weight than the MST $$T$$, which is a contradiction.