Answer: to convey that peace and strength will grow in all directions.
The Iroquois Confederacy included several different peoples, in particular, the Mohawk, Onondaga, Oneida, Cayuga, and Seneca. This multicultural nature encouraged imagery that conveys unity and peace. The Tree of Peace serves that purpose. It serves as a symbol of unity among people in different geographical locations. It also serves as an avenue to explain the benefits that will come to people for being part of the Confederacy, such as peace and strength. The phrases that convey this feeling are: "the spreading branches of the Tree of Peace" and "their nature [of the roots] is peace and strength."