Red-Black Tree

A red-black tree is a binary search tree that satisfies the following properties:

• Search key property: a red-black tree is a binary search tree.
• Color property: each node is either black or red.
• Root property: the root is black.
• Internal property: the children of a red node are black.
• Depth property: all the leaf nodes have the same black depth, which is defined as the number of black nodes along the branch from the root to the leaf node.

Intuitively, there is a mapping between a red-black tree and a (2,4)-tree.

Main operations:

• lookup: same as binary search tree lookup.
• insert:
  

  Insert the new node as a normal BST insert;
  If the new Node is the root
      color it black;
  Else
      color it red;
  If there is a double-red problem, i.e., the parent node is red
      Case 1: the sibling of the parent node is black
              Perform a tri-node restructure
      Case 2: the sibling of the parent node is also red
              (then the grand parent node must be black)
              color its parent node and parent's sibling node both black,
              color its grand parent node red, double red problem may propagate
  

  Performance: At most 1 tri-node restructure and Ο(logN) recoloring.
• remove:

  Find the victim node to delete
  If the victim node has two children
      swap it with its in-order successor node v // v has no left child
  Remove node v, place its right child r in its position
  // r can be null; and null pointers are all considered black
  If v is red, then its right child r must be black
      done
  If its right child r is red, v must be black
      color r black, done
  If v is black and r is also black
      color r double-black, and we have an underflow problem
      Case 1: sibling y of r is black and has a red child z
           perform a tri-node restructure, done
      Case 2: sibling y of r is black and both its children are black
           if y and r's parent node x is red
               color x black, color y red, color r black, done
           if x is black
               color y red, color r black, color x double black, 
               double black problem propagate
      Case 3: sibling y of r is red, y and r's parent must be black
           perform a rotation adjustment, and reduce this case to case 1 or case 2
  

  Performance: At most 1 tri-node restructure, 1 adjustment and Ο(logN) recoloring.