Answer:
In the study of America's history, the phrase "Roots of the Tree of Liberty" is first used by Jefferson in his letter to William Stephens Smith.
In that letter, Jefferson states that the "The tree of liberty must be occassionally statisfied with the blood of patriots and tyrants"
Scholars have posited that he was referring to the Constitution of the United States of America which is founded on the Declaration of Independence. The shedding of blood meant that people at some point would give their lives to defend the constitution.
It is worthy of note that Thomas Jefferson is credited with drafting the Declaration of Independence in 1776 and he did so under three days. It was amended and ratified on the 4th of July the same year.
At the root of this declaration is the "natural rights philosophy" that all people are entitled to “life, liberty, and the pursuit of happiness,” and that where a government does not protect such rights, the people have the choice to depose such a government.
It is therefore natural that when Jefferson became president in 1800 he pursued this ideal vigorously hence the reason why the root grew the fastest under his presidency.
Under his regime, in 1807, he criminalized international slave trade by signing The Act Prohibiting Importation of Slaves into law. This along with the purchase the entire Louisiana territory for $ 15 Million are recorded as some of his major achievements.
Cheers!