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Red-Black trees

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• A more efficient kind of balanced tree
  • The AVL tree is the classical balanced tree

  • Later, a more efficient form of balanced tree was invented - the Red-Black tree (Sedgewick showed that the insert operation can be implemented in just 46 lines of Java code.)

  • Wikipedia contains the history of its evolution: click here

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• Properties of Red-Black trees
  • A Red-Black tree is a Binary Search tree with the following (additional) properties:

    Each node in the binary search tree is labeled either as red or as black The root node is always labeled as black All the leaf nodes are always labeled as black Every red node must have 2 black children nodes (There is no child constraint on black (parent) nodes) Every path from root node → leaf node must have the same number of black nodes

    Example of a red-black tree: (from Wikipedia):

    (Every path from root node → leaf node in this Red-Black tree has exactly 3 black nodes)

    Notice:

    The number of red nodes on a path from root node → leaf node can be different !!!

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• Minimum and maximum path length in a Red-Black tree
  • Consider the properties #4 and #5 in a Red-Black tree:

    Each node in the binary search tree is labeled either as red or as black The root node is always labeled as black All the leaf nodes are always labeled as black Every red node must have 2 black children nodes (There is no child constraint on black (parent) nodes) Every path from root node → leaf node must have the same number of black nodes

    From property #4, we can conclude that:

    The child node of a red node must be a black node I.e.: a Red-Black tree cannot have 2 consecutive red nodes on a path !!! (Consecutive black nodes are allowed)

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  • Suppose a red-black tree has N black nodes on each path from root node → leaf node

    Then:

    The minimum (smallest) number of nodes of a path from root node → leaf node is equal to N:       A path from root node → leaf node cannot have any more black nodes (otherwise, the Red-black tree will violate property #5) A path from root node → leaf node can include more red nodes !!! Example: We have have at most N − 1 red nodes in a path from root node → leaf node: Therefore: the maximum (largest) number of nodes of a path from root node → leaf node is equal to N + (N−1) = 2N−1