TreesDefinitions• Read Weiss, 4.1 – 4.2

Implementation• Nodes and Links• One Arrays• Three Arrays

Traversals• Preorder, Inorder, Postorder

K-ary Trees• Converting trees: k-ary ↔ binary

Definitions

NodesEdgesRoot nodeInterior nodeLeafParentChildren

Ancestor / Proper ancestorDescendant / Proper

descendantLevel of a nodeHeight of a node / treeDegree of a node / treeDepth of nodePathAcyclic graph

bj

c

dg

k

m l

i h f e

a root, height=4, depth=level=0

leaf, height=0, depth=level=4

degree=2

interior node, height=2, depth=level=2

degree of tree = 2height of tree = 4

descendants of “a”

proper descendants of “a”

degree=0

degree=1

Implementing a Tree • Nodes and Links

A

B C

D

▲▲ ▲

▲ ▲

Node { Object value; Node lchild; Node rchild;} // Node

▲=null link

Implementing a Tree • One array

A

B C

D

▲▲ ▲

▲ ▲

A: 0 1 2 3 4 5 6 7 8 9

- A B C - - D - - -

▲=null link

• A[1] is root• lchild of A[i] is A[2i]• rchild of A[i] is A[2i+1]• “-” means array element is null / not used• A[0] not used as a node• A[0] may be used to hold general info (e.g., number of nodes in tree)

Preorder N L R

Traversals

preorder (Node t) if (t == null) return;

visit (t.value()); preorder (t.lchild()); preorder (t.rchild());

} // preorder

Traversalsinorder (Node t) if (t == null) return;

inorder (t.lchild()); visit (t.value()); inorder (t.rchild());

} // inorder

• Inorder L N R

Traversalspostorder (Node t) if (t == null) return;

postorder (t.lchild()); postorder (t.rchild()); visit (t.value());

} // postorder

• Postorder

L R N

bj

c

dg

k

m l

i h f e

a

preorder: a j k m l b c g i h d f e

inorder: m k l j a b i g h c f d e

postorder: m l k j i h g f e d c b a

K-ary Trees

n

g

e

jk

md

c

b

f

ip

q

a

degree of tree = 4degree of nodes f and n = 3height of tree = 3depth=level of m = 2

K-ary Tree => Binary Tree

n

g

e

jk

md

c

b

f

ip

q

a

K-ary Binaryroot rootleftmost child left childright sibling right child

n

g

e

jk

md

c

b

f

ip

q

a

n

g

e

jk

md

c

b

f

ip

q

a

TraversalsK-ary Tree Binary Tree

Preorder:Inorder:Postorder:

Preorder:Inorder:Postorder:

Binary Search Trees

Have all properties of binary treeItems in left subtree are smaller than items in

any nodeItems in right subtree are larger than items in

any node

BST Properties

Items must be comparableAll items have a unique valueGiven two distinct items x and y either

value(x) < value(y)value(x) > value(y)

If value(x) = value(y) then x = yIt will simplify programming to assume there

are no duplicates in our set of items.

Items

Need to map Items to a numerical valueIntegers

Value(x) = xPeople

Value(x) = ssnValue(x) = student id

Items

ConstructorInsertFind

FindminFindmax

Remove

BST Operations

Generally Recursive

BST Operations

BinaryNode operation( Comparable x, BinaryNode t ) {// End of path

if( t == null ) return null;

if( x.compareTo( t.element ) < 0 )return operation( x, t.left );

else if( x.compareTo( t.element ) > 0 )return operation( x, t.right );

else return t; // Match}

private BinaryNode find( Comparable x, BinaryNode t )

{ if( t == null ) return null; if( x.compareTo( t.element ) < 0 ) return find( x, t.left ); else if( x.compareTo( t.element ) > 0 ) return find( x, t.right ); else return t; // Match }

BST Find Method

RemoveNode is leaf

Remove nodeNode has one child

Replace node with childNode has two children

Replace node with smallest child of right subtree.

BST Remove Operations

private BinaryNode remove( Comparable x, BinaryNode t ) { if( t == null ) return t; // Item not found; do nothing if( x.compareTo( t.element ) < 0 ) t.left = remove( x, t.left ); else if( x.compareTo( t.element ) > 0 ) t.right = remove( x, t.right ); else if( t.left != null && t.right != null ) // Two children { t.element = findMin( t.right ).element; t.right = remove( t.element, t.right ); } else t = ( t.left != null ) ? t.left : t.right; return t; }

Remove method

Review depth/heightDepth

Depth is number of path segments from root to node

Depth of node is distance from root to that node.

Depth is uniqueDepth of root is 0

Internal Path Length

HeightHeight is maximum distance from node to a

leaf.There can be many paths from a node to a leaf.The height of the tree is another way of saying

height of the root.

Internal Path Length

IPL is the sum of the depths of all the nodes in a tree

It gives a measure of how well balanced the tree is.

Internal Path Length

Internal Path Length

11

2

N = 4IPL = 1 + 1 + 2 = 4

Internal Path Length

1

2

N = 4IPL = 1 + 2 + 3 = 6

3

Calculate IPL of all possible trees

Average IPL for N nodesN = 4

1

2

3

11

2

1

2 2

Simple to understandWorks for small datasetsBasis for more complicated treesUsing inheritance can implement

AVL treesSplay treesRed Black trees

Where do BST fit in