Notes October 31,2006

Happy Halloween from Kimberly Hart

TREES

Binary tree

Generally implemented using linked lists

Can be implemented as an array –

Implementation of a tree element (What would a node look like in this case?)

--Node and two pointers

--left pointer and a right pointer

Type link is new access node;

Type node is record

value : integer;

left : link;

right : link;

What might you want to do with the tree? i.e. what operations will have to be implemented?

--add values to the tree

--delete values from the tree

--search for a value in the tree

--Order the tree – generally done as you construct it!!!

~~--get parent, - not possible with our node structure~~

~~--add parent – not possible with our node structure~~

--Make empty

--Print the values in the tree starting at the root. – called TRAVERSING the tree.

Parts of the tree

root-—has no PARENT

every other node has at most two children—0,1,or two children.

a node with no children is a LEAF.

a node other than the root with 1 or 2 children is an internal node

TRAVERSING THE TREE

To go through tree level by level - Use a queue
BREADTH FIRST SEARCH— requires a level by level traversal of a tree

To go down a branch to the bottom is called a DEPTH FIRST traversa;
There are three such traversals

pre-order
in-order
post-order
all are implemented recursively

Pre-order

Visit a node (i.e. do something with its value – e.g. print it)

Traverse its left sub-tree

Traverse its right sub-tree

In-order

Traverse a node’s left sub-tree

Visit the node (i.e. do something with its value – e.g. print it)

Traverse the node’s right sub-tree

Post-order

Traverse the node’s left sub-tree

Traverse the node’s right sub-tree

Visit the node (i.e. do something with its value – e.g. print it)

IF we stored the tree in an array, here’s what the array structure would look like

Array position	1	2	3	4	5	6	7	Etc
	Root	Left child of root	Right child of root	Left child of left child of root	Right child of left child of root	Left child of right child of root	Right child of right child of root

Note that if a node is stored in position n, its children are stored in positions 2n and 2n+1

Other terms

--Binary tree means each node has at most two children

--Height –number of levels in a tree, start with 0

--Ancestor—someone further up the tree

--Descendant-below a given node in a tree

PERFECT BINARY TREE—a binary tree which all of the leaves are on the same level and every non leaf has two children.

HEAP—binary tree implemented as an array

binary tree definition – (our text – [age 630] )

A tree in which each node is capable of having two child nodes, a left child node and a right child node.

Each subtree is a binary tree

Binary tree versus—binary search tree --- Search tree is ordered!

		5
-2				7
	3		6		17

A binary search tree is a tree in which all of the values in the left sub-tree of the root are smaller than the value in the root and all of the values in the right sub-tree of the root are greater than the value in the root and one where each of the sub-trees of the root is a binary search tree.

Start of building a binary search tree

--check for empty tree

-- if empty—pointer points to new node which becomes root of tree (or subtree)

--else

--if the value in the node n greater than root.value go right

--else go left

At each point follow the same pattern

Pre-Order DEPTH FIRST SEARCH—go down

Check root

If not empty—print value then check left st. if not empty print value until hit empty subtree then back up… (to be discussed further in class)

If you look at the binary tree in figure 10.3 on page 629, here are the three traversals: