1. Trees, Binary Trees, and Binary Search Trees

2. Trees Linear access time of linked lists is prohibitive Trees
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

3. 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

4. Tree can be viewed as a ‘nested’ lists: Tree is also a graph …
list of lists of lists …
Tree is also a graph …

5. Some Terminologies Child and Parent Leaves Sibling
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

6. More Terminologies Path Length of a path Depth of a node
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

7. Example: UNIX Directory

8. Tree Operations Traversal, the most important
we will not implement a general ‘tree’, so wont discuss
Search
Insertion
Deletion

9. 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

10. 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 }

11. preorder(tree)
if empty(tree) stop
else
print(element(tree))
preorder(child(tree))
preorder(next(tree))

12. Example: Unix Directory Traversal
PreOrder
PostOrder

13. 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-case binary tree

14. 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 …

15. 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 }

16. 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

17. Preorder, Postorder and Inorder
Preorder traversal
node, left, right
prefix expression
++a*bc*+*defg

18. Preorder, Postorder and Inorder
Inorder traversal
left, node, right
infix expression
a+b*c+d*e+f*g
Postorder traversal
left, right, node
postfix expression
abc*+de*f+g*+

19. Preorder, Postorder and Inorder Pseudo Code

20. Binary tree insertion and deletion …

21. BST: Binary Search Tree

22. Recall the binary search algorithm4 7 10 15 19 20 42 54 87 90

23. 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

24. Binary Search Trees
A binary search tree
Not a binary search tree

25. Binary Search Trees Average depth of a node is O(log N)
Maximum depth of a node is O(N)
The same set of keys may have different BSTs

26. Searching BST If we are searching for 15, then we are done.
If we are searching for a key < 15, then we should search in the left subtree.
If we are searching for a key > 15, then we should search in the right subtree.

28. 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

29. 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

30. 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

31. 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)

32. void insert(double x, BinaryNode*& t){
if (t==NULL) t = new BinaryNode(x,NULL,NULL);
else if (x<t->element) insert(x,t->left);
else if (t->element<x) insert(x,t->right);
else ; // do nothing }

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

34. 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.

35. 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

36. 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)

37. void remove(double x, BinaryNode*& t){
if (t==NULL) return;
if (x<t->element) remove(x,t->left);
else if (t->element < x) remove (x, t->right);
else if (t->left != NULL && t->right != NULL) // two children
t->element = finMin(t->right) ->element;
remove(t->element,t->right); }
else
Binarynode* oldNode = t;
t = (t->left != NULL) ? t->left : t->right;
delete oldNode;

38. Deletion Example
Removing 40 from (a) results in (b) using the smallest element in the right subtree (i.e. the successor) 5 30 2 40 80 35 32 33 31 85 60 5 30 2 60 80 35 32 33 31 85
(a)
(b)

39. Removing 40 from (a) results in (c) using the largest element in the left subtree (i.e., the predecessor) 5 30 2 40 80 35 32 33 31 85 60 5 30 2 35 80 32 33 31 85 60
(a)
(c) 39

40. Removing 30 from (c), we may replace the element with either 5 (predecessor) or 31 (successor). If we choose 5, then (d) results. 5 30 2 35 80 32 33 31 85 60 2 5 35 80 32 33 31 85 60
(c)
(d) 40

41. Example: compute the number of nodes?

42. 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 42

43. 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 43

44. 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 would be the successor of x?
The parent of x!
Search trees 44

45. 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 45

46. Three Scenarios to Determine Successor
Successor(x)
x has right descendants
=> minimum( right(x) )
x has no right descendants
Scenario I
x is the left child of some node
=> parent(x)
x is the right child of some node
Scenario II
Scenario III
Search trees 46

47. 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 47

48. 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; }; 48

49. 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! 49

50. Example Successor(root, 13) Part I Traverse tree from root to find 13
order -> 15, 6, 7, 13 7 6 15
Stack s 50

51. 7 6 15 Stack s Successor(root, 13) Part II Find Parent (Scenario III)
y=s.pop()
while y!=NULL and x=right(y)
x = y;
if s.isempty()
y=NULL
else
return y
y =pop()=6
y =pop()=7
x = 13 7 6 15
Stack s 51

52. Make a binary or BST ADT …

53. 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;
Composition of a tree is not completely given by the constructors,
The composition is a level above ‘constructor’
(insert and remove are different from those of BST)

54. 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;
Composition of a tree is actually given by the constructors already
BST is for efficient search, insertion and removal, so restricting these functions.

55. 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;
Weiss textbook:
Trees with different implemntations are called differently,
For example, a heap is usually implemneted with an array

56. Comments: root, left subtree, right subtree are missing:
1. we can’t 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 it’s two layers, the public for BST, and the private for Binary Tree.
3. it might be defined internally in ‘private’ part (actually it’s implicitly done).
Trees with different implemntations are called differently,
For example, a heap is usually implemneted with an array

57. 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 (t==NULL) t = new BinaryNode(x,NULL,NULL);
else if (x<t->element) insert(x,t->left);
else if (t->element<x) insert(x,t->right);
else ; // do nothing }

58. By inheritance All search, insert and deletion have to be redefined.
Struct Node {
double element; // the data
Node* left; // left child
Node* right; // right child }
Class BinaryTree { …
class BST : public BinaryTree {
void BST::search () {
void BST::insert () {
void BST::delete () {
template<typename T>
Struct Node {
T element; // the data
Node* left; // left child
Node* right; // right child }
class BinaryTree { …
class BST : public BinaryTree<T> {
void BST<T>::search (const T& x) {
void BST<T>::insert (…) {
void BST<T>::delete (…) {
Composition of a tree is actually given by the constructors already
All search, insert and deletion have to be redefined.

59. More general BST …
template<typename T>
Struct Node {
T element; // the data
Node* left; // left child
Node* right; // right child }
class BinaryTree { …
class BST : public BinaryTree<T> {
void BST<T>::search (const T& x) {
void BST<T>::insert (…) {
void BST<T>::delete (…) {
template<typename T>
class BinaryTree { … }
template<typename T, typename K>
class BST : public BinaryTree<T> {
void BST<T>::search (const K& key) {
Composition of a tree is actually given by the constructors already
Search ‘key’ of K might be different from the data record of T!!!

60. Deletion Code (1/4)
First Element Search, and then Convert Case III, if any, to Case I or II
template<class E, class K>
BSTree<E,K>& BSTree<E,K>::Delete(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)
BinaryTreeNode<E> *p = root, // search pointer
*pp = 0; // parent of p
while (p && p->data != k){
// move to a child of p
pp = p;
if (k < p->data) p = p->LeftChild;
else p = p->RightChild; } 60

61. Deletion Code (2/4) if (!p) throw BadInput(); // no element with key k
e = p->data; // save element to delete
// restructure tree
// handle case when p has two children
if (p->LeftChild && p->RightChild) {
// two children convert to zero or one child case
// find predecessor, i.e., the largest element in // left subtree of p
BinaryTreeNode<E> *s = p->LeftChild,
*ps = p; // parent of s
while (s->RightChild) {
// move to larger element
ps = s;
s = s->RightChild; } 61

62. Deletion Code (3/4) // move from s to p p->data = s->data;
p = s; // move/reposition pointers for deletion
pp = ps; }
// p now has at most one child
// save child pointer to c for adoption
BinaryTreeNode<E> *c;
if (p->LeftChild) c = p->LeftChild;
else c = p->RightChild;
// deleting p
if (p == root) root = c; // a special case: delete root
else {
// is p left or right child of pp?
if (p == pp->LeftChild) pp->LeftChild = c;//adoption
else pp->RightChild = c; 62

63. Deletion Code (4/4)
delete p;
return *this; } 63