# (c) (6 points) Show a formal proof that the height of a Red-Black Tree h with... (c) (6 points) Show a formal proof that the height of a Red-Black Tree h with n nodes, is at most 2.log(n + 1), i.e. h < 2.log(n + 1). Show and proof any intermediate claim in order to prove the final claim.

Solution

A red black tree has the following properties:

1) Every node has a color either red or black.

2) Root of tree is always black.

3) A red node cannot have a red parent or red child.

4) Every path from a node to any of its descendant (NULL node) has the same number of black nodes.

For a binary tree, let k be the minimum number of nodes from root to a leaf node, then total number of nodes in the binary tree >=2k - 1

Nomber of nodes at root level (Level 1) = 1

Number of nodes in level 2 = 2

Number of nodes in level 2 = 22

Number of nodes in level 2 = 23

Number of nodes in level 2 = 24

Number of nodes in level 2 = 25

..................................................

Number of nodes in level k = 2k-1

So, minimmum number of nodes in binary tree = 1 + 2 + 22 + 23 + 24 + .................+ 2k-1

= 1.(2k - 1) / 2 - 1 = 2k - 1

n >= 2k​​​​​​​ - 1

n+1>=2k​​​​​​​

2k​​​​​​​ <= n+1

log22k = log (n+1)

k = log(n+1)

So a root to leaf path will have at-most log2(n+1) black nodes.

Since a red node cannot have a red parent or a red child, at least 1/2 of the nodes on any path from root to an external node are black. Since the longest path of the tree is h, the black-height of the root must be at least h/2

So for a Red-Black Tree of height h has black-height >= h/2.

h/2 <=   log2(n+1)

h <= 2.log2(n+1)

So the height of red black tree with n nodes is atmost 2.log2(n+1)

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