Red-Black tree is a binary search tree with following properties. a. Every node is either red or black. b. Every leaf node (nil) is black. c. If a node is red, then both of its children are black. d. All paths from a node to its descendant leaves contains the same number of black nodes. e. The root node is black. a) As we have learned, for Red-Black trees, the black-height of a node x, bh (x), is the number of black nodes (including the leaf node) on the path from x to any leaf, not counting x Prove that the subtree rooted at any node x contains greaterthanorequalto 2^bh (x) - 1 internal nodes. b) Prove that the insert operation of the red-black tree is always log (n) Red-Black tree is a binary search tree with following properties. a. Every node is either red or black. b. Every leaf node (nil) is black. c. If a node is red, then both of its children are black. d. All paths from a node to its descendant leaves contains the same number of black nodes. e. The root node is black. a) As we have learned, for Red-Black trees, the black-height of a node x, bh (x), is the number of black nodes (including the leaf node) on the path from x to any leaf, not counting x Prove that the subtree rooted at any node x contains greaterthanorequalto 2^bh (x) - 1 internal nodes. b) Prove that the insert operation of the red-black tree is always log (n)