## Binary Tree

A binary tree $$T$$ is a structure defined on a finite set of nodes that either

• contains no nodes, or
• is composed of three disjoint set of nodes: a root node, a binary tree called its left subtree, and a binary tree called its right subtree.

The binary tree that contains no nodes is called the empty tree or null tree, sometimes denoted NIL.

## Binary Search Tree

The keys in a binary search tree are always stored in such a way as to satisfy the binary-search-tree property:
Let $$x$$ be a node in a binary search tree. If $$y$$ is a node in the left subtree of $$x$$, then $$y.key \geq x.key$$. If $$y$$ is a node in the right subtree of $$x$$, then $$y.key \leq x.key$$.

#### Theorem 1

If $$x$$ is the root of an $$n$$-node subtree, then the call INORDER-TREE-WALK($$x$$) takes $$\Theta(n)$$ time.

### Querying a binary search tree

TREE-SEARCH(x, k)

if x == NIL or k == x.key
return x
if k < x.key
return TREE-SEARCH(x.left, k)
else return TREE-SEARCH(x.right, k)


ITERATIVE-TREE-SEARCH(x, k)

while x != NIL and k != x.key
if k < x.key
x = x.left
else x = x.right
return x


TREE-MINIMUM(x)

while x.left != NIL
x = x.left
return x


TREE-MAXIMUM(x)

while x.right != NIL
x = x.right
return x


#### Theorem 2

We can implement the dynamic-set operations SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR so that each one runs in $$O(h)$$ time on a binary search tree of height $$h$$.

#### Theorem 3

We can implement the dynamic-set operations INSERT and DELETE so that each one runs in $$O(h)$$ time on a binary search tree of height $$h$$.

#### Theorem 4

The expected height of a randomly built binary search tree on $$n$$ distinct keys is $$O(\lg{n})$$.

## Red-Black Tree

A red-black tree is a binary search tree with one extra bit of storage per node: its color, which can be either RED or BLACK.
A red-black tree is a binary tree that satisfies the following red-black properties:

• Every node is either red or black.
• The root is black.
• Every leaf (NIL) is black.
• If a node is red, then both its children are black.
• For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.

#### Lemma 1

A red-black tree with $$n$$ internal nodes has height at most $$2 \lg{(n+1)}$$.

## References

1. Introduction to Algorithms, MIT Press