Why a Pine Tree?
The pine tree had long been a symbol of the colonies, and had been used as a part of many flags dating back to the 1680s. Though not as well known as the Boston Tea Party, the Pine Tree Riot of 1772 was one of the more important acts of resistance by the American colonists leading up to the Revolutionary War. (The Pine Tree Riot was against a tax placed on colonists by the British in order to harvest certain pine trees.)
Why An Appeal To Heaven?
The phrase "An Appeal to Heaven" comes from John Locke, whose writings were extremely influential on the thinking of Thomas Jefferson and other leaders of the American Revolution. In particular, the phrase comes from Locke's Second Treatise on Civil Government. The phrase was much quoted among the colonists, including Patrick Henry in his "Give me Liberty, or give me death" speech.
The phrase refers to one of the justifications for armed resistance to tyrannical governments - that once all Earthly appeals (to the tyrannical government and its legal system) have been exhausted, an appeal to Heaven (God) may be made through resistance and revolution.
This idea is expressed in the Declaration of Independence, where the colonists state that their appeals to the British authorities failed, and therefore they were now "appealing to the Supreme Judge of the world" by declaring their independence.