1 foundations of software design fall 2002 marti hearst lecture 17: binary search trees; heaps

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Foundations of Software DesignFall 2002Marti Hearst

Lecture 17: Binary Search Trees; Heaps

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Our Binary Tree

• Defined recursively as consisting of– Data (in this example: int and String, but can be

anything!)– A lefthand Binary Tree– A righthand Binary Tree

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Simple Binary Tree Code

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Depth-first Search

From http://www.rci.rutgers.edu/~cfs/472_html/AI_SEARCH/ExhaustiveSearch.html

Full tree

DFS (to see it, run inPresentationMode)

Process first all nodes at the deepest levelof the tree

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Depth First Search

val

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Breadth-first Search

From http://www.rci.rutgers.edu/~cfs/472_html/AI_SEARCH/ExhaustiveSearch.html

Full tree

BFS (to see it, run inPresentationMode)

Process all nodes atdepth 1, then all thenodes at depth 2, and so on..

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Breadth First Traversal

We can reuse our Queue!!

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Binary Search Trees

• Differs from Binary trees in that– Each internal node has a key– The key is used to give an order to the nodes– Nodes often store data

• But sometimes stored only in leaf (B+Trees)– The keys are stored in order such that

• Keys to the left of a node are less than keys to the right of that node

– More formally,• For each internal node v with key k

– Keys stored in nodes in the left subtree of v are <= k– Keys stored in nodes in the right subtree are >= k

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Searching in a Binary Search Tree

• What is the running time?– Equivalent to the height of the tree O(h)– This can be O(n) in an unrestricted tree– We have ways to maintain the shape of the tree to

make it O(log n) (AVL trees)

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Updating the Binary Tree

• Inserting is pretty straightforward– Add node at the “bottom” of the tree– O(h)

• Deleting gets more complex– If the node w to be removed has two internal nodes

below it, • Find the first node that follows w via an inorder travesal• This is y, the leftmost internal node in the right subtree of w• Move the item stored at y into w’s position• Remove the now orphaned leaf that had been below y as

well as y– O(h)

• Nice applet:– http://www.seanet.com/users/arsen/avltree.html

13Slide copyright 1999 Addison Wesley Longman

A heap is a certain kind of complete binary tree.

When a completebinary tree is built,

its first node must bethe root.

When a completebinary tree is built,

its first node must bethe root.

RootHeaps


Left childof theroot

The second node isalways the left child

of the root.

The second node isalways the left child

of the root.


Right childof the

root

The third node isalways the right child

of the root.

The third node isalways the right child

of the root.


The next nodesalways fill the next

level from left-to-right..


level from left-to-right..



level from left-to-right.




Each node in a heapcontains a key thatcan be compared toother nodes' keys.

Each node in a heapcontains a key thatcan be compared toother nodes' keys.

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The "heap property"requires that each

node's key is >= thekeys of its children

The "heap property"requires that each

node's key is >= thekeys of its children

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Put the new node in the next available spot.

Move the new node upward, swapping with its parent until the new node reaches an acceptable location.

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Adding a Node to a Heap


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The parent has a key that is >= new node, or

The node reaches the root.

The process of moving the new node upward is called reheapification upward.

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Move the last node onto the root.

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Removing the Top of the Heap


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Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.

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The children all have keys <= the out-of-place node, or

The node reaches the leaf.

The process of pushing the new node downward is called reheapification downward.

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• Store the data from the nodes in a partially-filled array.

An array of dataAn array of data

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Implementing a Heap


Data from the root goes in the first location of the array.


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Data from the next row goes in the next two array locations.


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We don't care what's inWe don't care what's inthis part of the array.this part of the array.


• The links between the tree's nodes are not actually stored as pointers, or in any other way.

• The only way we "know" that "the array is a tree" is from the way we manipulate the data.


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If you know the index of a node, then it is easy to figure out the indexes of that node's parent and children. Formulas are given in the book.

[0][0] [1] [2] [3] [4]

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Next Time: Hash Tables

1 foundations of software design fall 2002 marti hearst lecture 17: binary search trees; heaps

Documents

slide copyright

righthand binary tree

nodes nodes

root heaps slide

simple binary tree code

search val slide

lefthand binary tree

binary tree inserting