I got an exercise about a variation of RB trees but I am struggling to see how to solve it, therefore I'll be happy to hear your opinion about it.
The exercise is:
Let us define a binary search tree which we will call Red-Red-Black tree if it satisfies the following:
- Every node is either red or black.
- Every leaf (NIL) is black.
- If a node is red, and its parent is red as well then both its children are black.
- For each node, all paths from the node to descendant leaves contain the same number of black nodes.
Find the largest possible number of internal nodes in Red-Red-Black tree with N nodes, such that their black height is $k$ find also what is the smallest possible number.
I was trying to think what I can deduce about the possible height of a node when I know its black height $k$.
So because of property number 3, we know that after each black node in the path we can have at most 2 consecutive red nodes, so I was wondering if I can say that the biggest possible height of a node that has a black height of $k$ is at most $3k$ and then try to figure out the cases about the possible level in the tree of nodes that has black height $k$ (Another observation which I am not sure about is: when we have a node which is not the root and this node has a black-height of $k$ then the siblings of this node must have the same black-height).
I will appreciate any clues. Thanks
