Why didn’t the Chinese leave a trip wire force in North Korea after the Korean War in a similar fashion to the United States leaving a trip wire force in South Korea?
The question should be asked backwards: why did the US use tripwire deterrence in South Korea?
Stationing troops in foreign soil should be considered unusual. In the case of tripwire deterrence, a nation deliberately places her troops in harm's way so that they are sacrificed when the host nation is attacked. The death of her own troops then creates a strong political will to retaliate. The US uses tripwire deterrence mainly in Central Europe, but also in South Korea.
This was not really necessary for China. China is an authoritarian regime that can more easily deploy her troops, politically speaking. During the Korean War, China intervened without having any of her troops killed or territory invaded first. The decision to intervene was based on UN encroachment in Korea and among members of China's Politburo.
There were two reasons: 1) Economics and 2) Geography.
The first was fairly obvious. Today, China is almost neck-and-neck with the U.S. as the world's largest economy, but in 1950, that was very different, with China then having only one-sixth of the U.S. GDP.
The second is that China is right across the border (Yalu River) from North Korea, while the U.S. is about 9000 miles from South Korea. In 1950, it took a full month for the U.S. to deploy forces in South Korea (except for General Dean's 24th Division based in Japan). In the event of an emergency, China can move forces "ninety" miles and have them in North Korea in a day or two.
China was a very, very poor country. More poor, than N.Korea, BTW. The Northern Korea was more industrialized than Southern Korea these days. So, even if China wanted to, it couldn't.
Have you heard that Kim Ir Sen asked China for that help? I haven't. The Chuchkhe paradigm was based on the isolation. So, nobody asked China for that, too.
China in 50-ties was the lesser brother of the USSR. It 4 times asked to join as a republic to it. It was USSR then who decided who will do what in N. Korea.
And Southern Korea heavily needed foreign forces to protect it.
The question is not why didn't the Chinese leave troops behind.
The question really is why would anyone want to leave troops behind in a foreign friendly country after a war? (Japan, Germany are not friendly nations. They are militaristic nations, that if left alone will likely develop weapons and military forces that can rival ours and threaten the world. so leaving forces there is perfectly logical).
It serves no strategic, economic, political purpose.
Maintaining an army overseas is expensive. In the event of war, an ally country typically has sufficient military strength to hold on until reinforcement arrives. In America's case, with America's mighty (and mighty expensive) carrier strike groups, America can deploy firepower that can match most top tier countries anywhere in the world within a couple of weeks. In China's case, they have plenty of rail links to North korea. If they are needed, it wouldn't take them all that long to get there either.
And it is hard to derive any political benefits, as a concentration of fit and energetic young men and women often cause unwanted disturbances for the locals.