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# Binary search tree

In computer science, a binary search tree is a binary tree where every node has a value, every node's left subtree has values less than the node's value, and every right subtree has values greater. Note that this requires a linear order on the values. A new node is added as a leaf. There is a sort algorithm based on binary search trees, and also a search algorithm. An in-order traversal of a binary search tree will visit the values in increasing order.

If we write our binary tree nodes as triples (left subtree, node, right subtree), and the null pointer as None, we can build and search them as follows (in Python):

```def binary_tree_insert(treenode, value):
if treenode is None: return (None, value, None)
left, nodevalue, right = treenode
if nodevalue > value:
return (binary_tree_insert(left, value), nodevalue, right)
else:
return (left, nodevalue, binary_tree_insert(right, value))
def build_binary_tree(values):
tree = None
for v in values:
tree = binary_tree_insert(tree, v)
return tree
def search_binary_tree(treenode, value):
if treenode is None: return None  # failure
left, nodevalue, right = treenode
if nodevalue > value:
return search_binary_tree(left, value)
elif value > nodevalue:
return search_binary_tree(right, value)
else:
return nodevalue
```

Note that the worst case of this build_binary_tree routine is O(n2) --- if you feed it a sorted list of values, it chains them into a linked list with no left subtrees. For example, build_binary_tree([1, 2, 3, 4, 5]) yields the tree (None, 1, (None, 2, (None, 3, (None, 4, (None, 5, None))))). There are a variety of schemes for overcoming this flaw with simple binary trees.

Once we have a binary tree in this form, a simple inorder traversal can give us the node values in sorted order:

```def traverse_binary_tree(treenode):
if treenode is None: return []
else:
left, value, right = treenode
return (traverse_binary_tree(left) + [value] + traverse_binary_tree(right))
```

So the binary tree sort algorithm is just the following:

```def treesort(array):
array[:] = traverse_binary_tree(build_binary_tree(array))
```

## Types of Binary Search Trees

There are many types of binary search trees. AVL trees and red-black trees are both forms of self-balancing binary search trees. A B-tree grows from the bottom up as elements are inserted. A splay tree is a self-adjusting binary search tree. In a treap ("tree heap"), each node also holds a priority and the parent node has higher priority than its children.