And in international relations, it's generally very much a case of give-an-inch-and-they'll-take-a-mile. Seemingly disproportionate importance must be given to the tiniest things, because the enemy will be constantly testing you looking for weakness.
In a case like this, if the US and South Korea aren't hyper-vigilant about every single thing, it's an indication that political or popular support may be waning, and an indication to North Korea that it might be possible to reinvade and take over the entire peninsula.
If you don't believe this, it's a big part of what allowed Germany to take over Europe in WWII -- that vigilance wasn't taken seriously. Germany pushed what was "acceptable" more and more until it was too late.
It's not about the tree -- it's about fundamental legitimacy being upheld.