1

A

B

ED

C

H

GF

I

J

LK

TREES• So far…. Data structures for linearly-ordered items

• Trees… “non-linear” arrangements of itemshierarchical organizationpart / wholeabove / belowinside / outsideancestor / descendant

…

L-17-__-19-20

[Portions © Goodrich & Tomassia]

COMP-2001 ~ Computer Science Department ~ University College Dublin ~ www.cs.ucd.ie/staff/nick ~ © Nicholas Kushmerick 2001

2

Trees• trees

• binary trees

• traversals of trees

• template method pattern

• data structures for trees

Lec 17

This week:Read Chapter 5Complete P7Start P8

3

Trees

• a tree represents a hierarchyexamples: organization structure of a corporation

table of contents of a book

Lec 17

4

Another Example• Unix or DOS/Windows file system

5

Terminology

6

Tree ADT• Trees make use of Position objects (like List & Sequence)

element()

• The Tree methods are:root() returns the Position for the rootparent(p) returns the Position of p’s parentchildren(p) returns an Iterator of the Positions

of p’s childrenisInternal(p) does p have children?isExternal(p) is p a leaf?isRoot(p) is p==root()?size() number of nodesisEmpty() always false!positions() returns an Iterator of every node

in some arbitrary order [more onspecific traversal orders in Lect 19!]

7

Binary Trees

8

Examples of Binary Trees

•arithmetic expression

•river

9

ADTs for Binary Trees• Just like a regular tree, but 4 additional operations

leftChild(p) return the Position of p’s left childrightChild(p) return the Position of p’s right childsibling(p) returns the other child of p’s parentexpandExternal(p) “grow” the binary tree by

converting p to a internal nodeand adding 2 new leaf nodes

(exceptions raised if the requestednode does not exist)

10

A unified hierarchy of 5 “Container” ADTs

InspectableVector InspectableList

InspectableSequenceVector List

Sequence[see G&T figures 5.11 and

6.11 for discussion]

Youare

here

Getting these interfaces “right” is acraft/art that takes years of experiences

11

Using the BinaryTree ADTBinaryTree t = new BinaryTree();

t.root().setElement(“Albert”);

t.expandExternal(tree.root());

t.root().leftChild().setElement(“Betty”);

t.root().rightChild().setElement(“Chris”);

t.expandExternal(tree.root().leftChild());

t.root().leftChild().leftChild().setElement(“David”);

t.root().leftChild().rightChild().setElement(“Elvis”);

“Albert”

“Chris”“Betty”

“Elvis”“David”

12

Trees… continuedLec 19

This week: Finish chapter 5complete P8start P9

13

Implementing BinaryTree• Trees are inherently “non-linear” so it might appear that

arrays can’t be used. Not quite…… • The trick: store value at position # i,

left child at # (2i), right child at # (2i+1)

+

+ 4

3 1- 2

9 3

1 2 3 4 5 6 7 8 9 10 11

14

Linked Data Structure for Binary Trees

15

A generalization of doubly-linked-lists

16Implementation details• class BTNode implements Position {

Object element;BTNode left, right, parent;BTNode(Object e, BTNode l, BTNode r, BTNode p) {

element=e; left=l; right=r; parent=p;}

}• class LinkedBinaryTree implements BinaryTree {

Position root;int size = 1;LinkedBinaryTree() { root = new BTNode(null,null,null.null);}boolean isInternal(Position p) { return ((BTNode)p).getLeft()!=null) && ((BTNode)p).getright()!=null);}Position leftChild(Position p) { return ((BTNode)p).getLeft();}boolean expandExternal(Position p) { ((BTNode)p).setLeft(new BTNode(null,(BTNode)p,null,null)); ((BTNode)p).setRight(new BTNode(null,(BTNode)p,null,null)); size += 2;}

...

[pages267-269

for details]

17

Tree operation #1 - depthThe depth of a node v in atree t is the number of edgesfrom the T’s root to v.

depth of Albert = 0depth of David = 2

“Albert”

“Chris”“Betty”

“Elvis”“David”

int depth(Tree t, Position p) {if (T.isRoot(p)) {

return 0;} else {

return 1 + depth(T, T.parent(p));}

}

18

DIGRESSION: Recursion• Functions can call themselves• The basic idea: express answer to some computation in

terms of same computation on “simpler” argument.• Must ensure recursion terminates eventually!• Some algorithms (eg, tree traversal!) are naturally

implemented recursively.

int factorial(int n) { if (n <= 1) { return 1; } else { return n * factorial(n-1); }}

non-recursive case toensure termination withsimplest arguments

recursive call

simpler argument

19

Writing a recursive function• Split the computation into several parts:

– some that have immediate answer• factorial(1) = 1

– and others that have answers expressible in terms of simpler results:

• factorial(n) = n * factorial(n-1)

• The “trick” is to ensure that:– the recursive call(s) must simplify the computation in some way

– must be special case(s) to handle simplest computation(s)

n! = n*(n-1)*(n-2)* …

*3*2*1

if somehow we could get

the answer to this...

… then we could get

the answer to this.

20

Recursion is very helpful for data structures that are composed of “parts” that are themselves instances of that same data structure.

The basic structure of all recursive tree operations:To do some operation on the tree, you just performthat operation on each of the tree’s children

More on recursion

a tree!

21

Tree Operation #2 - height

The height of a tree t is thenumber of edgesfrom the T’s root to v.

height of tree = 2

height of tree = height of Albert= 1 + height of Betty (ignore Chris because Betty has deeper children)

= 1 + height of David (could use Elvis instead)

= 0 = 1 = 2

“Albert”

“Chris”“Betty”

“Elvis”“David”

22

Height, continuedint height(Tree t) {

return height(T, T.root());}

int height(Tree t, Position p) {if (T.isExternal(p)) {

return 0;} else {

int h = 0;Iterator c = T.children(v);while (c.hasNext()) {

Position child = (Position) c.next();h = max(h, height(T,child));

}return 1 + h;

}}

23Lec 20

Trees… continued

24

Operations #3 - traversing a tree• Systematic way of “visiting” or “processing”

each node in the tree.• A very common operation:

– print out each node’s label– add 10 to each node’s label– examine each node to find label “Elvis”– count total number of nodes

…• Q: But in which order are nodes visited??• A: many “standard” orders:

– pre-order traversal– in-order traversal– post-order traversal– Euler traversal– breadth-first traversal (all nodes at depth=0, then depth=1, then depth=2)

…

Cover these now

breadth first: COMP-2006 will discuss “graphs”

25

Additions to the BinaryTree ADT

• traversal functions– Iteration inOrder();– Iteration preOrder();– Iteration postOrder();– Iteration eulerOrder();

– Returns the Positions of every nodein the specified order, as an Iterator

• Note that Tree.positions [slide 6] is alsoa traversal -- but order not specified

26

Traversing Trees - #1 - Pre-Order• preorder traversal

Algorithm preOrder(v)“visit” node vfor each child w of v do

recursively perform preOrder(w)

• EXAMPLE - readinga document frombeginning to end

Don’t reada section

until read allprior sections

27

Implementation of preOrder• Uses Java’s “Vector” to collect nodes

in the correct order•

Iteration preOrder() {Vector nodes = new Vector();preOrder(root(), nodes); // collect nodesreturn nodes.iterator(); // ‘convert’ Vector to Iterator

}void preOrder(Position p, Vector nodes) {

nodes.addElement(p); // visit this nodeif (isExternal(p)) return; // stop at leavespreOrder(leftChild(p), nodes);preOrder(rightChild(p), nodes);

}

28

Using preOrder• Example: pre-order print-out of a tree’s labels

void preOrderPrint(BinaryTree tree) {for (Iterator i = tree.preOrder(); e.hasMore(); ) {

Position p = (Position) i.next; System.out.println(p.element()); }}

Position find(BinaryTree tree, Object value) {for (Iterator i = tree.preOrder(); e.hasMore(); ) {

Position p = (Position) i.next; if (p.element.equals(value)) return p; }

return null; // not found!}

29

Traversing Trees #2 - Post-Order• postorder traversal

Algorithm postOrder(v)

for each child w of v do

recursively perform postOrder(w)

“visit” node v

• du (disk usage) command in Unix

to computevalue for parent,

must first visitall children

30

Traversing Trees - #3 - In-Order• inorder traversal of a binary tree (inapplicable to non-binary trees)

algorithm inOrder(v) recursively perform inOrder(leftChild(v)) “visit” node v recursively perform inOrder(rightChild(v))

•printing an arithmetic expressionspecialization of an inorder traversalprint “(“ before traversing the left subtreeprint “)” after traversing the right subtree

print operatorbetweenchildren

31

Traversal #4 - Euler Tour Traversal• generic traversal of a binary tree

• the preorder, inorder, and postorder traversals are special cases of the Euler tour traversal

• “walk around” the tree and visit each node three times:– on the left

– from below

– on the right

32Comparison: All 4 Traversal AlgorithmspreOrder:1. visit this node2. preOrder left child3. preOrder right childinOrder:1. inOrder left child2. visit this node3. inOrder right childpostOrder:1. postOrder left child2. postOrder right child3. visit this node

only differenceis when thisstep occurs

Euler:1. visit this node2. postOrder left child3. Visit this node4. postOrder right child5. visit this node

33

QuestionsWhat is the preOrder traversal of this tree?

A

B

ED

C

H

GF

I

J

LK

34

QuestionsWhat is the inOrder traversal of this tree?

A

B

ED

C

H

GF

I

J

LK

35

QuestionsWhat is the postOrder traversal of this tree?

A

B

ED

C

H

GF

I

J

LK

36

QuestionsWhat is the Euler traversal of this tree?

A

B

ED

C

H

GF

I

J

LK