Do red black trees have different rules and shapes?

Question

I followed the rules of the insert operation listed at GeeksforGeeks. I used them to solve example 2 here. My solution below is different than the one in the same document.

                32 (black)
           /                \
         21 (black)          64 (black)
        /                     \
      15 (red)                 75 (red)

My questions are:

Are there many sets of rules for rotations and recoloring? Should the red black tree be unique for a sequence of insert operations for a set of numbers, using different rules? Is the tree in my solution considered "balanced"?

P.s. to clarify, what is different is that when I insert a node to a red node with a black grandparent that has a red sibling, I should recolor the parent and the sibling black, and recolor the grandparent red. If that creates two reds, then the recoloring is propagated until it reaches the root, which should be colored black.

Answer 1

Red-black trees are characterized by a set of invariants. As long as those invariants are satisfied, you have a legal red-black tree. There is no uniqueness property. In fact, the second solution shown in the link you posted is exactly your solution, which is indeed balanced.

Some implementations only produce trees where red nodes cannot be right children of their parents. That produces yet another red-black tree for your example that differs from yours in that a black 75 is the right child of the root and has a red 64 as left child.