Christofides algorithm: why must an MST have even number of odd-degree vertices?

Question

This question is not necessarily related to Christofides algorithm per se, I just ran into it when reading about it.

I assume that a minimum spanning tree must have an even number of odd-degree vertices, since Christofides algorithm uses this fact to find a perfect matching between "the set of vertices with odd degree in $T$" - an impossible mission for a graph with odd number of vertices...

I couldn't convince myself why this is true, can someone please convince me? Either by a formal proof or with an informal explanation.