How does one determine when a war is won or lost? what is the purpose of a treaty?

The purpose of a treaty is to officially end the state of war between the hostile parties.

You can win or lose a war in many ways. It depends on the time in history. A nation can surrender after enough losses, give the territories to the winner or the whole country in some cases. But capitulation is not the only way to win a war. You can lose every single part of land a country owns but continue to fight with your allies like Serbia did in World War 1. And Germany surrendered in World War 1 not by losing territory but by realizing that there is no way to win the war. And in World War 2 they were fighting until the end. Even after the capitulation, some soldiers kept fighting.

So to win a war you need the other side to surrender. Casualties, territory, and length of war do not mean victory or defeat, only when one party concedes defeat.

answer; i believe the correct answer is (the willamete valley);

increase in population