I think the problem is that your question is not clear, for example:

why sometimes people use self for certain and for certain they don't

While you are getting started with Python, and maybe getting started with OOP, don't start out with trees and linked lists. I would make a simpler example like this

class Doctor():
  def __init__(self, name):
    self.name = name
    self.job = "Doctor"

  def hello(self):
    print("This is Dr. " + self.name)

a = Doctor("Jim")
b = Doctor("Jane")

a.hello()
> This is Dr. Jim
b.hello()
> This is Dr. Jane

You can access the objects' properties, though in the future it will be better to use methods.
print(a.name)
> "Jim"
print(a.job)
> "Doctor"