Ah, but it was as it was one of the points in your counter-argument to which you stated:
"No tree or plant, at any point in its life, develops sentience."
By the verbiage you used to illustrate your claim, you state that sentience is part of the criteria for something to be "alive." This argument you have made actually supports my claim that a fetus is not "alive" until the fetus itself reaches sentience.
Having said that... Regarding your claim on what a "person" is...
A person (plural people or persons) is a being that has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of property, or legal responsibility. The defining features of personhood and consequently what makes a person count as a person differ widely among cultures and contexts.
Personhood is the status of being a person. Defining personhood is a controversial topic in philosophy and law, and is closely tied to legal and political concepts of citizenship, equality, and liberty. According to common worldwide general legal practice, only a natural person or legal personality has rights, protections, privileges, responsibilities, and legal liability. Personhood continues to be a topic of international debate, and has been questioned during the abolition of slavery and the fight for women's rights, in debates about abortion, fetal rights, and in animal rights advocacy.
-Richard A. Shweder/Edmund J. Bourne. 1982. Does the Concept of the Person Vary Cross-Culturally?, in: Anthony J. Marsella/Geoffrey M. White (eds), Cultural Conceptions of Mental Health and Therapy, Dordrecht, S. 97-137.
So, the definition of personhood is subjective as it is based on personal values and thus cannot be 100% defined as a universally recognized truth. Whereas there is no debate about what sentience is or what the requirements for it are.
Based upon the loose definition of personhood, and the absolute definition of sentience, I have drawn my logical conclusion on when a fetus is a person. To be a person, you require sentience in some capacity - the ability to feel physically or emotionally at its most core levels. That is the minimum criteria to be a person. Yet, there are other arguments that follow that line of reasoning as animals can also feel pain. They can also reason. So, it's not so simple.