Left-leaning red–black tree
A left-leaning red–black (LLRB) tree is a type of self-balancing binary search tree, introduced by Robert Sedgewick. It is a variant of the red–black tree and guarantees the same asymptotic complexity for operations, but is designed to be easier to implement.

Properties 
A left-leaning red-black tree satisfies all the properties of a red-black tree:

 Every node is either red or black.
 A NIL node is considered black.
 A red node does not have a red child.
 Every path from a given node to any of its descendant NIL nodes goes through the same number of black nodes.
 The root is black (by convention).
Additionally, the left-leaning property states that:
  If a node has only one red child, it must be the left child.
The left-leaning property reduces the number of cases that must be considered when implementation search tree operations.

Relation to 2–3 and 2–3–4 trees 

LLRB trees are isomorphic 2–3–4 trees. Unlike conventional red-black trees, the 3-nodes always lean left, making this relationship a 1 to 1 correspondence. This means that for every LLRB tree, there is a unique corresponding 2–3–4 tree, and vice versa.

If we impose the additional requirement that a node may not have two red children, LLRB trees become isomorphic to 2–3 trees, since 4-nodes are now prohibited. Sedgewick remarks that the implementations of LLRB 2–3 trees and LLRB 2–3–4 trees differ only in the position of a single line of code.

Analysis 
All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of  in a tree of  keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close to the optimal  nodes examined that would be observed in a perfectly balanced tree.

Specifically, in a left-leaning red-black 2–3 tree built from  random keys, Sedgewick's experiments suggest that:
 A random successful search examines  nodes.
 The average tree height is about 
 The average size of left subtree exhibits log-oscillating behavior.

Bibliography 
 Robert Sedgewick's Java implementation of LLRB from his 2008 paper
 Robert Sedgewick. 20 Apr 2008. Animations of LLRB operations
 Open Data Structures - Section 9.2.2 - Left-Leaning Red–Black Trees, Pat Morin

References

External links 
 Robert Sedgewick. Left-leaning Red–Black Trees. Direct link to PDF.
 Robert Sedgewick. Left-Leaning Red–Black Trees slides from October 2008.
 Linus Ek, Ola Holmström and Stevan Andjelkovic. May 19, 2009. Formalizing Arne Andersson trees and Left-leaning Red–Black trees in Agda
 Julien Oster. March 22, 2011. An Agda implementation of deletion in Left-leaning Red–Black trees
 Kazu Yamamoto. 2011.10.19. Purely Functional Left-Leaning Red–Black Trees
 Left-Leaning Red-Black Trees Considered Harmful

Search trees