Trees 1© 2004 Goodrich, Tamassia

Trees

Make Money Fast!

StockFraud

PonziScheme

BankRobbery

Trees 2© 2004 Goodrich, Tamassia

What is a TreeIn computer science, atree is an abstract modelof a hierarchicalstructureA tree consists of nodeswith a parent-childrelationApplications: Organization charts File systems Programming

environments

Computers”R”Us

Sales R&DManufacturing

Laptops DesktopsUS International

Europe Asia Canada

Trees 3© 2004 Goodrich, Tamassia

subtree

Tree TerminologyRoot:Internal node:

External node (a.k.a. leaf ):

Ancestors of a node:

Depth of a node:

Height of a tree:

Descendant of a node:

A

B DC

G HE F

I J K

Subtree:Root: node without parent (A)Internal node: node with at leastone child (A, B, C, F)External node (a.k.a. leaf ): nodewithout children (E, I, J, K, G, H, D)Ancestors of a node: parent,grandparent, grand-grandparent,etc.Depth of a node: number ofancestorsHeight of a tree: maximum depthof any node (3)Descendant of a node: child,grandchild, grand-grandchild, etc.

Subtree: tree consisting ofa node and itsdescendants

Trees 4© 2004 Goodrich, Tamassia

Preorder TraversalA traversal visits the nodes of atree in a systematic mannerIn a preorder traversal, a node isvisited before its descendantsApplication: print a structureddocument

Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme1.1 Greed 1.2 Avidity

2.3 BankRobbery

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Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)

Trees 5© 2004 Goodrich, Tamassia

Postorder TraversalIn a postorder traversal, anode is visited after itsdescendantsApplication: compute spaceused by files in a directory andits subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/todo.txt

1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

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Trees 6© 2004 Goodrich, Tamassia

Binary TreesA binary tree is a tree with thefollowing properties: Each internal node has at most two

children (exactly two for properbinary trees)

The children of a node are anordered pair

We call the children of an internalnode left child and right childAlternative recursive definition: abinary tree is either a tree consisting of a single node, or a tree whose root has an ordered

pair of children, each of which is abinary tree

Applications: arithmetic expressions decision processes searching

A

B C

F GD E

H I

Trees 7© 2004 Goodrich, Tamassia

Arithmetic Expression TreeBinary tree associated with an arithmetic expression internal nodes: operators external nodes: operands

Example: arithmetic expression tree for theexpression (2 × (a − 1) + (3 × b))

+

××

−2

a 1

3 b

Trees 8© 2004 Goodrich, Tamassia

Decision TreeBinary tree associated with a decision process internal nodes: questions with yes/no answer external nodes: decisions

Example: dining decision

Want a fast meal?

How about coffee? On expense account?

Starbucks Quizno’s Tarpy’s Café Del Monte

Yes No

Yes No Yes No

Trees 9© 2004 Goodrich, Tamassia

Properties of Proper Binary TreesNotationn number of nodese number of

external nodesi number of internal

nodesh height

Properties: h + 1 ≤ n ≤ 2h+1 − 1 1 ≤ e ≤ 2h

h ≤ i ≤ 2h − 1

log2 (n + 1) - 1 ≤ h ≤ (n + 1) / 2

Trees 10© 2004 Goodrich, Tamassia

Inorder TraversalIn an inorder traversal anode is visited after its leftsubtree and before its rightsubtreeApplication: draw a binarytree x(v) = inorder rank of v y(v) = depth of v

Algorithm inOrder(v)if hasLeft (v)

inOrder (left (v))visit(v)if hasRight (v)

inOrder (right (v))

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Trees 11© 2004 Goodrich, Tamassia

Print Arithmetic ExpressionsSpecialization of an inordertraversal print operand or operator

when visiting node print “(“ before traversing left

subtree print “)“ after traversing right

subtree

Algorithm printExpression(v)if hasLeft (v)

print(“(’’)inOrder (left(v))

print(v.element ())if hasRight (v)

inOrder (right(v))print (“)’’)

+

××

−2

a 1

3 b((2 × (a − 1)) + (3 × b))

Trees 12© 2004 Goodrich, Tamassia

Evaluate Arithmetic ExpressionsSpecialization of a postordertraversal recursive method returning

the value of a subtree when visiting an internal

node, combine the valuesof the subtrees

Algorithm evalExpr(v)if isExternal (v)

return v.element ()else

x ← evalExpr(leftChild (v))y ← evalExpr(rightChild (v))◊ ← operator stored at vreturn x ◊ y+

××

−2

5 1

3 2

Trees 13© 2004 Goodrich, Tamassia

Euler Tour TraversalGeneric traversal of a binary treeIncludes a special cases the preorder, postorder and inorder traversalsWalk around the tree and visit each node three times: on the left (preorder) from below (inorder) on the right (postorder)

+

×

−2

5 1

3 2

LB

R×

Trees 14© 2004 Goodrich, Tamassia

∅

Linked Structure for TreesA node is representedby an object storing Element Parent node Sequence of children

nodes

B

DA

C E

F

B

∅ ∅

A D F

∅

C

∅

E

Trees 15© 2004 Goodrich, Tamassia

Linked Structure for Binary TreesA node is representedby an object storing Element Parent node Left child node Right child node

B

DA

C E

∅ ∅

∅ ∅ ∅ ∅

B

A D

C E

∅

Trees 16© 2004 Goodrich, Tamassia

Array-Based Representation ofBinary Trees

nodes are stored in an array

…

let rank(node) be defined as follows: rank(root) = 1 if node is the left child of parent(node),

rank(node) = 2*rank(parent(node)) if node is the right child of parent(node),

rank(node) = 2*rank(parent(node))+1

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HG

FE

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J