Trees Terminology. A tree is a collection of elements (nodes) Each node may have 0 or more successors

Transcription

1 Trees Terminology A tree is a collection of elements (nodes) Each node may have 0 or more successors Except for one node, called the root, each node has exactly one predecessor Links from node to its successors are called branches Successors of a node are called its children Predecessor of a node is called its parent Nodes with same parent are siblings Nodes with no children are called leaves Also use words like ancestor and descendant

2 Example Tree

3 Trees Terminology A subtree of a node is a tree whose root is a child of that node The depth of a node is a measure of its distance from the root. Depth of the root is 0 Depth of other nodes is 1 + depth of parent The height of a node is the length of the path to its furthest child.

4 Example Tree Node Depth dog 0 cat 1 wolf 1 canine 2

5 Binary Trees In a binary tree a node has at most 2 non-empty subtrees A set of nodes T is a binary tree if either of these is true: T is empty The root of T has two subtrees, both binary trees Notice that this is a recursive definition

6 Expression Trees Non-leaf, internal, nodes contain operators Leaf nodes contain operands

7 Building Expression Trees

8 Huffman Trees Represents Huffman codes for characters appearing in a file or stream Huffman code may use different numbers of bits to encode different characters ASCII or Unicode uses a fixed number of bits for each character

9 Huffman Trees Letter Code b w s 0011 e 010

10 Binary Search Trees Binary search tree properties: Is a binary tree Elements in left subtree < element in subtree root Elements in right subtree > element in subtree root Search algorithm: if the tree is empty, return NULL if target equals to root node s data, return that data if target < root node data, return search(left subtree) otherwise return search(right subtree)

11 Fullness In computer science trees grow from the top down New values inserted in new leaf nodes In a full tree, every node has 0 or 2 non-null children

12 Completeness A complete tree of height h is filled up to depth h 1, and, at depth h, any unfilled nodes are on the right. A binary tree is complete if: All leaves are at level h or level h 1 (for some h) All level h 1 leaves are to the right

13 General Trees Nodes can have any number of children

14 First Child Next Sibling Representation General trees can be represented using a binary tree where the left link is to the first child and the right link is to the next sibling.

15 Traversals of Binary Trees Often want iterate over and process nodes of a tree.this process is called tree traversal. Three kinds of binary tree traversal: Preorder Inorder Postorder According to order of subtree root with respect to its children.

16 Traversals of Binary Trees Preorder: Visit root, traverse left, traverse right Inorder: Traverse left, visit root, traverse right Postorder: Traverse left, traverse right, visit root Preorder Traversal if the tree is empty return visit root traverse left subtree traverse right subtree Inorder Traversal if the tree is empty return traverse left subtree visit root traverse right subtree Postorder Traversal if the tree is empty return traverse left subtree traverse right subtree visit root

17 Visualizing Traversals You can visualize traversals by imagining an ant that walks along outside the tree If the ant keeps the tree on its left, it traces a route called the Euler tour Preorder traversal: record node first time the ant is there (ant passes on the way down) Inorder traversal: record after the ant traverses the left subtree (ant passes on the way to the right) Postorder traversal: record node the last time the ant is there (ant passes on the way up)

18 Visualizing Traversals

19 Traversals of Binary Search Trees In an inorder traversal of a binary search tree the nodes are visited in order of increasing data value Inorder traversal visits in the order: canine, cat, dog, wolf

20 Traversals of Expression Trees Inorder traversal can insert parentheses where they belong for infix form Postorder traversal results in postfix form Prefix traversal results in prefix form

21 BTNode Class Like linked list, a node has data and links to successors Data is reference to object of some type (i.e. BTNode is a template class) Binary tree node has links for left and right subtrees

22 Binary Tree Class

23 Binary Search Tree Class

24 Binary Search Tree Example

25 Binary Search Tree Search What is the algorithm for searching?

26 Binary Search Tree Insertion if the root is null replace the empty tree with a new tree with the item at the root return true else if item == root.data return false else if item < root.data insert on left subtree else insert on right subtree

27 Deletion in a Binary Search Tree How can we delete kissed? How can we delete is?

28 Deletion in a Binary Search Tree How can we delete kissed? kissed is at a leaf. Leaves can can just be removed. How can we delete is?

29 Deletion in a Binary Search Tree How can we delete kissed? kissed is at a leaf. Leaves can can just be removed. How can we delete is? is has only one child. In this case the child is used to replace its parent.

30 Deletion in a Binary Search Tree How can we delete rat?

31 Deletion in a Binary Search Tree How can we delete rat? rat has two children. In this case it is replaced by its inorder predecessor.

32 Deletion in a Binary Search Tree

33 Priority Queues A Priority Queue is an ADT with the following operations: removemin Removes and returns the element with the smallest key in the queue insert(x) Adds x to the queue Simple Implementations removemin insert Unsorted List Sorted List

34 Priority Queues A Priority Queue is an ADT with the following operations: removemin Removes and returns the element with the smallest key in the queue insert(x) Adds x to the queue Simple Implementations removemin insert Unsorted List O(n) O(1) Sorted List

35 Priority Queues A Priority Queue is an ADT with the following operations: removemin Removes and returns the element with the smallest key in the queue insert(x) Adds x to the queue Simple Implementations removemin insert Unsorted List O(n) O(1) Sorted List O(1) O(n)

36 Priority Queues A Priority Queue is an ADT with the following operations: removemin Removes and returns the element with the smallest key in the queue insert(x) Adds x to the queue Simple Implementations removemin insert Unsorted List O(n) O(1) Sorted List O(1) O(n) Our goal is to get both operations to be O(log(n)).

37 Heaps Heap properties: Is a complete binary tree The value at the root is the smallest item in the tree Every subtree is also a heap So in a heap the value at a node is less than (or equal) to all the values in its two subtrees.

38 Heaps As heaps are complete binary tree they are often stored in an array or ArrayList. Index: Key: parent(i) = (i + 1)/2 1 left(i) = 2i + 1 right(i) = 2i + 2

39 Size of a Complete Tree A complete tree, T, with n nodes has height log(n). Suppose the height of T is h...

40 Inserting a Key into a Heap insert(a, key): heapsize(a)++ i = heapsize(a) while i > 0 and A[parent(i)] > key: A[i] = A[parent(i)] i = parent(i) A[i] = key What is the time complexity of insert?

41 Inserting a Key into a Heap insert(a, key): heapsize(a)++ i = heapsize(a) while i > 0 and A[parent(i)] > key: A[i] = A[parent(i)] i = parent(i) A[i] = key What is the time complexity of insert? O(h) = O(log(n))

42 Heapify When a heap is manipulated we need to be sure to maintain the heap properties. heapify(a, i) assumes the trees rooted at left(i) and right(i) are heaps, but A[i] may violate the heap properties. So we will move A[i] down to make the tree rooted at i a heap.

43 Heapify heapify(a, i): l = left(i) r = right(i) if l < heapsize(a) and A[l] < A[i]: smallest = l else: smallest = i if r < heapsize(a) and A[r] < A[smallest]: smallest = r if smallest!= i: swap A[i] and A[smallest] heapify(a, smallest)

44 Heapify What is the time complexity of Heapify?

45 Heapify What is the time complexity of Heapify? O(h) = O(log(n))

46 removemin removemin(a): min = A[0] heapsize(a)-- heapify(a,0) return min What is the time complexity?

47 removemin removemin(a): min = A[0] heapsize(a)-- heapify(a,0) return min What is the time complexity? O(h) = O(log(n))

48 Building Heaps While we could use insert to build heaps, we can use heapify to build heap in a bottom up fashion. Given an array of length n if we think of the array as a complete tree indexes n/2 to n 1 are all of the leaves. Leaves are one elements heaps so we can use heapify to join them. buildheap(a) heapsize(a) = length(a) for i = length(a)/2-1 to 0: heapify(a,i) What is the time complexity?

49 Building Heaps While we could use insert to build heaps, we can use heapify to build heap in a bottom up fashion. Given an array of length n if we think of the array as a complete tree indexes n/2 to n 1 are all of the leaves. Leaves are one elements heaps so we can use heapify to join them. buildheap(a) heapsize(a) = length(a) for i = length(a)/2-1 to 0: heapify(a,i) What is the time complexity? Looks to be O(n log(n)). However, we can prove that it is O(n).

50 HeapSort With a heap it is easy to sort. heapsort(a) List sortedlist buildheap(a) while heapsize(a) > 0: sortedlist.add(removemin(a)) return sortedlist However this destroys A and uses extra memory for the sort.

51 HeapSort heapsort(a) buildheap(a) for i = length(a) - 1 down to 1: swap A[0] and A[i] heapsize(a)-- heapify(a, i) if A is a minheap: reverse(a) What is the time complexity?

52 HeapSort heapsort(a) buildheap(a) for i = length(a) - 1 down to 1: swap A[0] and A[i] heapsize(a)-- heapify(a, i) if A is a minheap: reverse(a) What is the time complexity? O(n log(n))