Why did european nations compete with each other to build colonies in north america

Because of resources. The more colonies there were, the more resources and land the nation could have for itself. Not only would you have access to natural resources such as gold or something similar, but also to other things such as animals for hunting which is something that the french colonies were famous for since they traded fur mostly. More resources equals more wealth and more wealth equals more power.

They made a checks and balance system to equal power within the branches. the "check" meant that one branch would watch one branch and "balance" them if they were too powerful. if you want more info on this, i would just google checks and balance system.