Trees, Binary Trees, and Binary Search Trees

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Trees, Binary Trees, and Binary Search Trees
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Trees

• Linear access time of linked lists is prohibitive
• Does there exist any simple data structure for
  which the running time of most operations
  (search, insert, delete) is O(log N)?
• Trees
• Basic concepts
• Tree traversal
• Binary tree
• Binary search tree and its operations

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Trees

• A tree T is a collection of nodes
• T can be empty
• (recursive definition) If not empty, a tree T
  consists of
• a (distinguished) node r (the root),
• and zero or more nonempty sub-trees T1, T2,
  ...., Tk

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• Tree can be viewed as a nested lists
• list of lists of lists
• Tree is also a graph

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Some Terminologies

• Child and Parent
• Every node except the root has one parent 
• A node can have a zero or more children
• Leaves
• Leaves are nodes with no children
• Sibling
• nodes with same parent

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More Terminologies

• Path
• A sequence of edges
• Length of a path
• number of edges on the path
• Depth of a node
• length of the unique path from the root to that
  node
• Height of a node
• length of the longest path from that node to a
  leaf
• all leaves are at height 0
• The height of a tree the height of the root
  the depth of the deepest leaf
• Ancestor and descendant
• If there is a path from n1 to n2
• n1 is an ancestor of n2, n2 is a descendant of n1
• Proper ancestor and proper descendant

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Example UNIX Directory
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Tree Operations

• Traversal, the most important
• we will not implement a general tree, so wont
  discuss
• Search
• Insertion
• Deletion

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Tree Traversal

• Used to print out the data in a tree in a certain
  order
• Pre-order traversal
• Print the data at the root
• Recursively print out all data in the leftmost
  subtree
• Recursively print out all data in the rightmost
  subtree

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A drawing of tree with two pointers
Struct TreeNode double element // the data
TreeNode child // FIRST child go to the
next generation TreeNode next // next SIBLING
go to the same generation
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preorder(tree) if empty(tree) stop else
print(element(tree)) preorder(child(tree))
preorder(next(tree))
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Example Unix Directory Traversal
PreOrder
PostOrder
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Binary Trees

• A tree in which no node can have more than two
  children
• The depth of an average binary tree is
  considerably smaller than N, even though in the
  worst case, the depth can be as large as N 1.

Generic binary tree
Worst-casebinary tree
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Binary Tree ADT

• Implementation
• at most two children, we can keep direct pointers
  to them
• a linked list is physically a pointer, so is a
  tree!
• Possible operations on the Binary Tree ADT
• Parent, left_child, right_child, sibling, root,
  etc
• Tree operations
• Search, insertion and deletion
• Make a Binary Tree ADT later

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A drawing of linked list with one pointer
A drawing of binary tree with two pointers
Struct BinaryNode double element // the data
BinaryNode left // left child BinaryNode
right // right child
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Example Expression Trees

• Leaves are operands (constants or variables)
• The internal nodes contain operators
• Will not be a binary tree if some operators are
  not binary

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Preorder, Postorder and Inorder

• Preorder traversal
• node, left, right
• prefix expression
• abcdefg

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Preorder, Postorder and Inorder

• Inorder traversal
• left, node, right
• infix expression
• abcdefg

• Postorder traversal
• left, right, node
• postfix expression
• abcdefg

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Preorder, Postorder and Inorder Pseudo Code
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Binary tree insertion and deletion
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BST Binary Search Tree
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Recall the binary search algorithm

•  

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Binary Search Trees (BST)

• A data structure for efficient searching,
  inser-tion and deletion
• Binary search tree property
• For every node X
• All the keys in its left subtree are smaller
  than the key value in X
• All the keys in its right subtree are larger
  than the key value in X

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Binary Search Trees

A binary search tree
Not a binary search tree
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Binary Search Trees
The same set of keys may have different BSTs

• Average depth of a node is O(log N)
• Maximum depth of a node is O(N)

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Searching BST

• If we are searching for 15, then we are done.
• If we are searching for a key lt 15, then we
  should search in the left subtree.
• If we are searching for a key gt 15, then we
  should search in the right subtree.

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Searching (Find)

• Find X return a pointer to the node that has key
  X, or NULL if there is no such node
• Time complexity O(height of the tree)

find(const double x, BinaryNode t) const
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Inorder Traversal of BST

• Inorder traversal of BST prints out all the keys
  in sorted order

Inorder 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20
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findMin/ findMax

• Goal return the node containing the smallest
  (largest) key in the tree
• Algorithm Start at the root and go left (right)
  as long as there is a left (right) child. The
  stopping point is the smallest (largest) element
• Time complexity O(height of the tree)

BinaryNode findMin(BinaryNode t) const
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Insertion

• Proceed down the tree as you would with a find
• If X is found, do nothing (or update something)
• Otherwise, insert X at the last spot on the path
  traversed
• Time complexity O(height of the tree)

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void insert(double x, BinaryNode t) if
(tNULL) t new BinaryNode(x,NULL,NULL) else
if (xltt-gtelement) insert(x,t-gtleft) else if
(t-gtelementltx) insert(x,t-gtright) else // do
nothing
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Insertion Example

• Construct a BST successively from a sequence of
  data
• 35,60,2,80,40,85,32,33,31,5,30

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Deletion

• When we delete a node, we need to consider how we
  take care of the children of the deleted node.
• This has to be done such that the property of the
  search tree is maintained.

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Deletion under Different Cases

• Case 1 the node is a leaf
• Delete it immediately
• Case 2 the node has one child
• Adjust a pointer from the parent to bypass that
  node

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Deletion Case 3

• Case 3 the node has 2 children
• Replace the key of that node with the minimum
  element at the right subtree
• (or replace the key by the maximum at the left
  subtree!)
• Delete that minimum element
• Has either no child or only right child because
  if it has a left child, that left child would be
  smaller and would have been chosen. So invoke
  case 1 or 2.
• Time complexity O(height of the tree)

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void remove(double x, BinaryNode t) if
(tNULL) return if (xltt-gtelement)
remove(x,t-gtleft) else if (t-gtelement lt x)
remove (x, t-gtright) else if (t-gtleft ! NULL
t-gtright ! NULL) // two children t-gteleme
nt finMin(t-gtright) -gtelement remove(t-gteleme
nt,t-gtright) else Binarynode oldNode
t t (t-gtleft ! NULL) ? t-gtleft
t-gtright delete oldNode
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Deletion Example

• Removing 40 from (a) results in (b) using the
  smallest element in the right subtree (i.e. the
  successor)

(b)
(a)
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• Removing 40 from (a) results in (c) using the
  largest element in the left subtree (i.e., the
  predecessor)

(c)
(a)
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40

• Removing 30 from (c), we may replace the element
  with either 5 (predecessor) or 31 (successor). If
  we choose 5, then (d) results.

(d)
(c)
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Example compute the number of nodes?
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Example Successor

• The successor of a node x is
• defined as
• The node y, whose key(y) is the successor of
  key(x) in sorted order
• sorted order of this tree.
  (2,3,4,6,7,9,13,15,17,18,20)

Some examples Which node is the successor of
2? Which node is the successor of 9? Which node
is the successor of 13? Which node is the
successor of 20? Null
Search trees
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Scenario I Node x Has a Right Subtree
By definition of BST, all items greater than x
are in this right sub-tree. Successor is the
minimum( right( x ) )
maybe null
Search trees
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Scenario II Node x Has No Right Subtree and x is
the Left Child of Parent (x)
Successor is parent( x ) Why? The successor is
the node whose key would appear in the next
sorted order. Think about traversal in-order.
Who wouldbe the successor of x? The parent of
x!
Search trees
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Scenario III Node x Has No Right Subtree and Is
Not a Left-Child of an Immediate Parent
Keep moving up the tree until you find a parent
which branches from the left().
Successor of x
y
Stated in Pseudo code.
x
Search trees
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Three Scenarios to Determine Successor
Successor(x)
x has right descendants gt minimum( right(x) )
x has no right descendants
Scenario I
x is the left child of some node gt parent(x)
x is the right child of some node
Scenario II
Scenario III
Search trees
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Successor Pseudo-Codes
Verify this code with this tree. Find successor
of 3 ? 4 9 ? 13 13 ? 15 18 ? 20
Note that parent( root ) NULL
Scenario I
Scenario II
Scenario III
Search trees
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Problem

• If we use a doubly linked tree, finding parent
  is easy.
• But usually, we implement the tree using only
  pointers to the left and right node. ? So,
  finding the parent is tricky.
• For this implementation we need to store the
  path ? Stack!

class Node int data Node left Node
right Node parent
class Node int data Node left Node
right
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Use a Stack to Find Successor
PART I Initialize an empty Stack s. Start at the
root node, and traverse the tree until we find
the node x. Push all visited nodes onto the
stack.
PART II Once node x is found, find
successor using 3 scenarios mentioned
before. Parent nodes are found by popping the
stack!
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Example
Successor(root, 13) Part I Traverse tree from
root to find 13 order -gt 15, 6, 7, 13
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6
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Stack s
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Successor(root, 13) Part II Find Parent
(Scenario III) ys.pop() while y!NULL
and xright(y) x y if s.isempty()
yNULL else ys.pop() return
y
y pop()6
y pop()7
x 13
7
6
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Stack s
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Make a binary or BST ADT
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For a generic (binary) tree
Struct Node double element // the data
Node left // left child Node right //
right child class Tree public Tree()
//
constructor Tree(const Tree t) Tree()
//
destructor bool empty() const double root()
// decomposition (access functions) Tree
left() Tree right() bool search(const
double x) void insert(const double x) //
compose x into a tree void remove(const double
x) // decompose x from a tree private Node
root
(insert and remove are different from those of
BST)
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For BST tree
Struct Node double element // the data
Node left // left child Node right //
right child class BST public BST()
//
constructor BST(const Tree t) BST()
//
destructor bool empty() const double root()
// decomposition (access functions) BST
left() BST right() bool serch(const double
x) // search an element void insert(const
double x) // compose x into a tree void
remove(const double x) // decompose x from a
tree private Node root
BST is for efficient search, insertion and
removal, so restricting these functions.
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Weiss textbook
class BST public BST() BST(const Tree
t) BST() bool empty() const bool
search(const double x) // contains void
insert(const double x) // compose x into a
tree void remove(const double x) // decompose x
from a tree private Struct Node double
element Node left Node right Node()
// constructuro for Node Node
root void insert(const double x, Node t)
const // recursive function void
remove() Node findMin(Node t) void
makeEmpty(Node t) // recursive
destructor bool contains(const double x, Node
t) const
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Comments
root, left subtree, right subtree are
missing 1. we cant write other tree
algorithms, is implementation dependent,
BUT, 2. this is only for BST (we only need
search, insert and remove, may not need other
tree algorithms) so its two layers, the
public for BST, and the private for Binary
Tree. 3. it might be defined internally in
private part (actually its implicitly done).
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A public non-recursive member function
void insert(double x) insert(x,root)
A private recursive member function
void insert(double x, BinaryNode t) if
(tNULL) t new BinaryNode(x,NULL,NULL) else
if (xltt-gtelement) insert(x,t-gtleft) else if
(t-gtelementltx) insert(x,t-gtright) else // do
nothing
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By inheritance
Struct Node double element // the data
Node left // left child Node right //
right child Class BinaryTree class BST
public BinaryTree void BSTsearch ()
void BSTinsert () void BSTdelete ()

templatelttypename Tgt Struct Node T element //
the data Node left // left child Node
right // right child templatelttypename
Tgt class BinaryTree templatelttypename
Tgt class BST public BinaryTreeltTgt void
BSTltTgtsearch (const T x) void
BSTltTgtinsert () void BSTltTgtdelete ()

All search, insert and deletion have to be
redefined.
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More general BST
templatelttypename Tgt Struct Node T element //
the data Node left // left child Node
right // right child templatelttypename
Tgt class BinaryTree templatelttypename
Tgt class BST public BinaryTreeltTgt void
BSTltTgtsearch (const T x) void
BSTltTgtinsert () void BSTltTgtdelete ()

templatelttypename Tgt class BinaryTree
templatelttypename T, typename Kgt class
BST public BinaryTreeltTgt void
BSTltTgtsearch (const K key)
Search key of K might be different from the
data record of T!!!
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Deletion Code (1/4)

• First Element Search, and then Convert Case III,
  if any, to Case I or II

templateltclass E, class Kgt BSTreeltE,Kgt
BSTreeltE,KgtDelete(const K k, E e) // Delete
element with key k and put it in e. // set p to
point to node with key k (to be
deleted) BinaryTreeNodeltEgt p root, // search
pointer pp 0 // parent of p while (p
p-gtdata ! k) // move to a child of p pp
p if (k lt p-gtdata) p p-gtLeftChild else p
p-gtRightChild
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Deletion Code (2/4)
if (!p) throw BadInput() // no element with key
k e p-gtdata // save element to delete //
restructure tree // handle case when p has two
children if (p-gtLeftChild p-gtRightChild) //
two children convert to zero or one child case //
find predecessor, i.e., the largest element in //
left subtree of p BinaryTreeNodeltEgt s
p-gtLeftChild, ps p // parent of s while
(s-gtRightChild) // move to larger element ps
s s s-gtRightChild
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Deletion Code (3/4)
// move from s to p p-gtdata s-gtdata p s
// move/reposition pointers for deletion pp
ps // p now has at most one child // save
child pointer to c for adoption BinaryTreeNodeltEgt
c if (p-gtLeftChild) c p-gtLeftChild else c
p-gtRightChild // deleting p if (p root)
root c // a special case delete root else
// is p left or right child of pp? if (p
pp-gtLeftChild) pp-gtLeftChild c//adoption else
pp-gtRightChild c
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Deletion Code (4/4)
delete p return this
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