Is it true that the catalyst was a conflict about a tree right on the border? I heard it from a teacher at a US university as he was teaching a course about nation-states in and out of war.
I think what your teacher may have been referring to is not the start of the Korean War in 1950, but the later axe murder incident, a serious border incident in 1976 which involved the deaths of two U.S. soldiers.
Following KIMH's magisterial Korean War, republished in English by a US academic press:
The Korean War began when alternative anti-Japanese (or in the Southern case, some pro-Japanese) factions of Korean nationalists [and some socialists] fell out and aligned with the respective great powers occupying their country. Both groups of nationalists wished to reunify Korea, as did the groundswell of working class socialism in Korea at the time. The questions were: would Korea have a working class revolution, and if it didn't or if it was brutally put down, would Soviet or American aligned imperialist running dog lackeys dominate Korea in their own interests?
From the Southern Perspective, their alliance with the United States was problematic. The United States was willing to equip a 10 division defensive army. Additionally the South faced a guerilla and industrial campaign by socialist activists on the ground (aligned and allied with the North). Additionally the Southern clique was disunified and incompetent.
From the Northern perspective, their alliance with the Soviet Union was problematic. The North wanted 3 or more armoured divisions, heavy air support, etc. They got one (I believe, recall?) which was more sensible, but the Soviet Union did outfit them with an offensive infantry army of about 10 divisions. The North also did not have to contend with workers' uprisings because the ideology of the Korean Bolshevik factions allowed them to hegemonise workers' discontent and bend it to their will. Additionally the Northern clique was unified (despite 4 factions plus the working class) and competent.
While elections for reunification had been promised, it is not surprising that the United States stymied these. What is surprising is that the United States stymied the unification elections out of gross incompetence, this is different from the normal case of US manipulation of such situations in the Cold War. The United States normally had a much more unified foreign policy.
The North asked permission to go to war when the strategy of southern uprising had failed, yet seemingly believed that an uprising was imminent (and could take Seoul from the inside). Permission for the war was given at a higher level due to ongoing tensions.
I highly recommend the English edition of the Korean Institute of Military History's history—it has been revised since greater democracy came to the South and is not too unsympathetic to the North.
Arguably, the Korean War was started when the then Secretary of State, Dean Acheson, drew a U.S. "defense perimeter" through the Sea of Japan, leaving South Korea outside it.
That may have caused North Korea's allies, Stalin's Soviet Union, and Mao Tse-tung's China, to give North Korea the "go ahead" to invade South Korea.