1. Data Structures and Algorithms
Trees
The definitions for this presentation are from from: Corman , et. al., Introduction to Algorithms (MIT Press), Chapter 5.
Some material on binomial trees is from Hull.
Data Structures and Algorithms

2. Data Structures and Algorithms
A Few Applications
Arithmetic Expressions
b + a * b + b * b a
Data Structures and Algorithms

3. Data Structures and Algorithms
A Few Applications
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XML Document
Object Model
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Data Structures and Algorithms

4. Binomial “Trees” Movements in Time dt
A Few Applications
Binomial “Trees” Movements in Time dt
Binomial trees are
frequently used to
approximate the
movements in the
price of a stock or
other asset
In each small interval
of time the stock price
is assumed to move up
by a proportional amount
u or to move down by a
proportional amount d. p Su S
1 – p Sd
From Hull

5. From O’Reilly “Mastering Bitcoin
A Few Applications
A Merkle tree – each block in a block chain
From O’Reilly “Mastering Bitcoin

6. Data Structures and Algorithms
Definitions
A Free tree is a connected, acyclic undirected graph.
Data Structures and Algorithms

7. Data Structures and Algorithms
If an undirected graph is acyclic but possibly disconnected,
it is a forest.
Data Structures and Algorithms

8. Data Structures and Algorithms
This is a graph that contains a cycle and is therefore neither a
tree nor a forest.
Data Structures and Algorithms

9. Theorem (Properties of Free Trees)
Let G = (V, E) be an undirected graph.
The following statements are equivalent:
1. G is a free tree.
Data Structures and Algorithms

10. Data Structures and Algorithms
2. Any two vertices in G are connected by a unique simple
path.
Data Structures and Algorithms

11. Data Structures and Algorithms
3. G is connected, but if any edge is removed from E, the
resulting graph is disconnected.
Data Structures and Algorithms

12. Data Structures and Algorithms
4. G is connected, and |E| = |V| - 1.
Data Structures and Algorithms

13. Data Structures and Algorithms
5. G is acyclic, and |E| = |V| - 1.
Data Structures and Algorithms

14. Data Structures and Algorithms
6. G is acyclic, but if any edge is added to E, the resulting
graph contains a cycle.
Data Structures and Algorithms

15. Data Structures and Algorithms
A rooted tree is a free tree in which one of the vertices is
distinguished from the others. The distinguished vertex is
called the root of the tree. We often refer to a vertex of a
rooted tree as a node (we may also call this a vertex) of the
tree. The following figure shows a rooted tree on a set of 12
nodes with root 7. 7 3 10 4 8 12 11 2 6 5 1 9
Data Structures and Algorithms

16. Data Structures and Algorithms
Consider a node x in a rooted tree T with root r. Any
node y on the unique path from r to x is called an ancestor
of x. If y is an ancestor of x, then x is a descendant of y.
(Every node is both an ancestor and a descendant of itself.)
If y is an ancestor of x and x  y, then y is a proper ancestor
of x and x is a proper descendant of y. The subtree rooted
at x is the tree induced by descendants of x, rooted at x. r 8 y 5 6 x 9
Data Structures and Algorithms

17. Data Structures and Algorithms
If the last edge on the path from the root r of a tree T to a
node x is (y, x), then y is the parent of x, and x is a child of
y. The root is the only node in T with no parent. If two
nodes have the same parent, they are siblings. A node with
no children is an external node or leaf. A nonleaf node is
an internal node.
Data Structures and Algorithms

18. Data Structures and Algorithms
The number of children of a node x in a rooted tree T is
called the degree of x. The length of the path from the
root r to a node x is the depth of x in T. The largest depth
of any node in T is the height of T. 7
depth 0
depth 1 3 10 4
height = 4 8 12 11 2
depth 2 6 5 1
depth 3 9
depth 4
Data Structures and Algorithms

19. Data Structures and Algorithms
An ordered tree is a rooted tree in which the children of
each node are ordered. That is, if a node has k children,
then there is a first child, a second child, …, and a kth
child. The two trees shown below are different when
considered to be ordered trees, but the same when
considered to be just rooted trees. 7 7 4 3 10 4 3 10 12 8 11 2 11 2 8 12 1 6 5 6 5 1 9 9
Data Structures and Algorithms

20. Data Structures and Algorithms
A binary tree T is a structure defined on a finite set of
nodes that either
contains no nodes, or
is comprised of three disjoint sets of nodes:
a root node, a binary tree called its left subtree
and a binary tree called its right subtree. 3 2 7 4 1 5 6
Data Structures and Algorithms

21. Data Structures and Algorithms
Full binary tree: each node is either a leaf or has degree
exactly 2.
In a positional tree, the children of a node are labeled
with distinct positive integers. The ith child of a node is
absent if no child is labeled with integer i. A k-ary tree
is a positional tree in which for every node, all children
with labels greater than k are missing. Thus, a binary
tree is a k-ary tree with k = 2.
Data Structures and Algorithms

22. Data Structures and Algorithms
A complete k-ary tree is a k-ary tree in which all leaves
have the same depth and all internal nodes have degree k.
depth 0
depth 1
depth 2
height = 3
depth 3
Data Structures and Algorithms

23. Data Structures and Algorithms
How many leaves L does a complete binary tree of height h
have?
d = 0
d = 1
d = 2
The number of leaves at depth d = 2d
If the height of the tree is h it has 2h leaves.
L = 2h.
Data Structures and Algorithms

24. Data Structures and Algorithms
What is the height h of a complete binary tree with L leaves?
leaves = height = 0
leaves = height = 1
leaves = height = 2
leaves = L height = Log2L
Since L = 2h
Log2L = Log22h
h = Log2L
Data Structures and Algorithms

25. Data Structures and Algorithms
The number of internal nodes of a complete binary
tree of height h is ?
Internal nodes = height = 0
Internal nodes = height = 1
Internal nodes = height = 2
Internal nodes = height = 3
h-1 =  2i
= 2h - 1
2 - 1
Thus, a complete binary tree of height = h has 2h-1 internal
nodes.
Geometric series
Data Structures and Algorithms

26. Data Structures and Algorithms
The number of nodes n of a complete binary
tree of height h is ?
nodes = height = 0
nodes = height = 1
nodes = height = 2
nodes = 2h height = h
Since L = 2h
and since the number of internal nodes = 2h-1 the
total number of nodes n = 2h+ 2h-1 = 2(2h) – 1 = 2h+1- 1.
Data Structures and Algorithms

27. Data Structures and Algorithms
If the number of nodes is n then what is the height?
nodes = height = 0
nodes = height = 1
nodes = height = 2
nodes = n height = Log2(n+1) - 1
Since n = 2h+1-1
n + 1 = 2h+1
Log2(n+1) = Log2 2h+1
Log2(n+1) = h+1
h = Log2(n+1) - 1
Data Structures and Algorithms

28. Data Structures and Algorithms
Catalan Numbers
= (2n)!
(n+1) n! (2n-n)!
1,1,2,5,14,...
Data Structures and Algorithms

29. The number of distinct binary trees with n nodes
...
Data Structures and Algorithms

30. Data Structures and Algorithms
Class for Binary Nodes
public class BTNode {
private Object data;
private BTNode left;
private BTNode right;
...
Data Structures and Algorithms

31. Data Structures and Algorithms
public BTNode(Object obj,
BTNode l,
BTNode r) {
data = obj; left = l; right= r; }
public boolean isLeaf()
return (left == null) &&
(right == null);
...
Data Structures and Algorithms

32. Data Structures and Algorithms
Copying Trees
public static BTNode
treeCopy(BTNode t) {
if (t == null) return null;
else
BTNode leftCopy =
treeCopy(t.left);
BTNode rightCopy =
treeCopy(t.right);
return new BTNode(t.data,
leftCopy,
rightCopy); }
Data Structures and Algorithms

33. Data Structures and Algorithms
Tree Traversals
Preorder
Inorder
Postorder
Levelorder
Data Structures and Algorithms

34. Data Structures and Algorithms
public void preOrderPrint(){
System.out.println(data);
if (left != null)
left.preOrderPrint();
if (right != null)
right.preOrderPrint(); }
a b c d e f g a e b g c d f
Root, Left, Right
Data Structures and Algorithms

35. Data Structures and Algorithms
public void inOrderPrint(){
if (left != null) {
left.inOrderPrint()
System.out.println(data);
if (right != null)
right.inOrderPrint() }
c b d a f e g a
b e
c d f g
Left, Root, Right
Data Structures and Algorithms

36. Data Structures and Algorithms
public void postOrder(){
if (left != null
left.postOrder()
if (right != null)
right.postOrder()
System.out.println(data); }
c d b f g e a a
b e
c d f g
Left, right, root
Data Structures and Algorithms

37. Data Structures and Algorithms
levelorder (T) {
Q = makeEmptyQueue()
enqueue (T,Q)
until isempty (Q) {
p = dequeue(Q)
visit (p)
for each child  of P, in order, do
enqueue (, Q) }
a b e c d f g a
b e
c d f g
Data Structures and Algorithms

38. An Array Representation
Suppose we are lucky enough to be working with complete
binary trees.
We can store the tree in an array.
Let the root be at index 0 and let the left and right children
of node i be at indexes 2i+1 and 2i+2 respectively.
Lewis and Denemberg, Page 111
Data Structures and Algorithms

39. An Array Representation
isLeaf(i) : 2i + 1 >= n
leftChild(i) : 2i + 1 (none if 2i+ 1 >= n)
rightChild(i): 2i + 2 (none if 2i + 2 >= n)
leftSibling(i): i - 1 (none if i == 0 or i is odd)
rightSibling(i) : i + 1 (none if i = n-1 or i is even)
parent(i) = int((i-1)/2) (none if i == 0)
Works if the tree is “almost complete”, growing top to bottom
and left to right.
Lewis and Denemberg, Page 111
Data Structures and Algorithms

40. Binomial “Trees” Using an Array Representation
S0d
S0d
S0d 2
S0d 2
The array element bt[0] will be S0.
In general, given a node with index i at depth d,
its left child is located at bt[i + d+1] and
its right child is located at bt[i + d+2]
S0d 3
S0d 4
Data Structures and Algorithms