The word that does not belong is "trench warfare." The other three terms—appeasement, militarism, and alliance—are related to the political and military strategies that shaped international relations and conflicts, particularly during the lead-up to World War I and its aftermath.
"Appeasement" refers to the diplomatic policy of making concessions to an aggressive power to avoid conflict, which was notably practiced by European leaders in the 1930s before World War II. "Militarism" is the belief in building up strong armed forces to prepare for war and often involves a glorification of military power. An "alliance" is a formal agreement between two or more nations to cooperate for specific purposes, often in a military context.
In contrast, "trench warfare" is a specific military tactic that became prominent during World War I, characterized by soldiers fighting from deep trenches, leading to a stalemate on the battlefield. While trench warfare was a significant aspect of the war itself, it does not pertain to the broader political and ideological concepts represented by the other three words. Therefore, trench warfare is the term that does not fit with the others in this context.