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Binary Trees, Binary Search Trees Binary Trees, Binary Search Trees www.cs.ust.hk/~huamin/ COMP171/bst.ppt Trees • Linear access time of linked lists is prohibitive – Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? Trees • A tree is a collection of nodes – The collection can be empty – (recursive definition) If not empty, a tree consists of a distinguished node r (the root), and zero or more nonempty subtrees T1, T2, ...., Tk, each of whose roots are connected by a directed edge from r Some Terminologies • Child and parent – Every node except the root has one parent – A node can have an arbitrary number of children • Leaves – Nodes with no children • Sibling – nodes with same parent Some Terminologies • Path • Length – number of edges on the path • Depth of a node – length of the unique path from the root to that node – The depth of a tree is equal to the depth of the deepest leaf • Height of a node – length of the longest path from that node to a leaf – all leaves are at height 0 – The height of a tree is equal to the height of the root • Ancestor and descendant – Proper ancestor and proper descendant Example: UNIX Directory Binary Trees • A tree in which no node can have more than two children • The depth of an “average” binary tree is considerably smaller than N, eventhough in the worst case, the depth can be as large as N – 1. Example: Expression Trees • Leaves are operands (constants or variables) • The other nodes (internal nodes) contain operators • Will not be a binary tree if some operators are not binary Tree traversal • Used to print out the data in a tree in a certain order • Pre-order traversal – Print the data at the root – Recursively print out all data in the left subtree – Recursively print out all data in the right subtree Preorder, Postorder and Inorder • Preorder traversal – node, left, right – prefix expression • ++a*bc*+*defg Preorder, Postorder and Inorder • Postorder traversal – left, right, node – postfix expression • abc*+de*f+g*+ • Inorder traversal – left, node, right. – infix expression • a+b*c+d*e+f*g • Preorder • Postorder Preorder, Postorder and Inorder Binary Trees • Possible operations on the Binary Tree ADT – parent – left_child, right_child – sibling – root, etc • Implementation – Because a binary tree has at most two children, we can keep direct pointers to them compare: Implementation of a general tree Binary Search Trees • Stores keys in the nodes in a way so that searching, insertion and deletion can be done efficiently. Binary search tree property – For every node X, all the keys in its left subtree are smaller than the key value in X, and all the keys in its right subtree are larger than the key value in X Binary Search Trees A binary search tree Not a binary search tree Binary search trees Two binary search trees representing the same set: • Average depth of a node is O(log N); maximum depth of a node is O(N) Implementation Searching BST • If we are searching for 15, then we are done. • If we are searching for a key < 15, then we should search in the left subtree. • If we are searching for a key > 15, then we should search in the right subtree. Searching (Find) • Find X: return a pointer to the node that has key X, or NULL if there is no such node • Time complexity – O(height of the tree) Inorder traversal of BST • Print out all the keys in sorted order Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20 findMin/ findMax • Return the node containing the smallest element in the tree • Start at the root and go left as long as there is a left child. The stopping point is the smallest element • Similarly for findMax • Time complexity = O(height of the tree) insert • Proceed down the tree as you would with a find • If X is found, do nothing (or update something) • Otherwise, insert X at the last spot on the path traversed • Time complexity = O(height of the tree) delete • When we delete a node, we need to consider how we take care of the children of the deleted node. – This has to be done such that the property of the search tree is maintained. delete Three cases: (1) the node is a leaf – Delete it immediately (2) the node has one child – Adjust a pointer from the parent to bypass that node delete (3) the node has 2 children – replace the key of that node with the minimum element at the right subtree – delete the minimum element • Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2. • Time complexity = O(height of the tree) Priority Queues – Binary Heaps homes.cs.washington.edu/ ~anderson/iucee/Slides_326.../ Heaps.ppt 30 Recall Queues • FIFO: First-In, First-Out • Some contexts where this seems right? • Some contexts where some things should be allowed to skip ahead in the line? 31 Queues that Allow Line Jumping • Need a new ADT • Operations: Insert an Item, Remove the “Best” Item 6 2 15 23 insert 12 18 deleteMin 45 3 7 32 Priority Queue ADT 1. PQueue data : collection of data with priority 2. PQueue operations – insert – deleteMin 3. PQueue property: for two elements in the queue, x and y, if x has a lower priority value than y, x will be deleted before y 33 Applications of the Priority Queue • Select print jobs in order of decreasing length • Forward packets on routers in order of urgency • Select most frequent symbols for compression • Sort numbers, picking minimum first • Anything greedy 34 Potential Implementations insert deleteMin Unsorted list (Array) O(1) O(n) Unsorted list (Linked-List) O(1) O(n) Sorted list (Array) O(n) O(1)* Sorted list (Linked-List) O(n) O(1) 35 Recall From Lists, Queues, Stacks • Use an ADT that corresponds to your needs • The right ADT is efficient, while an overly general ADT provides functionality you aren’t using, but are paying for anyways • Heaps provide O(log n) worst case for both insert and deleteMin, O(1) average insert 36 Binary Heap Properties 1. Structure Property 2. Ordering Property 37 Tree Review Tree T A root(T): leaves(T): B C children(B): parent(H): D E F G siblings(E): H I ancestors(F): descendents(G): subtree(C): J K L M N 38 More Tree Terminology Tree T depth(B): A height(G): B C degree(B): D E F G branching factor(T): H I J K L M N 39 Brief interlude: Some Definitions: A Perfect binary tree – A binary tree with all leaf nodes at the same depth. All internal nodes have 2 children. height h h+1 11 2 – 1 nodes 2h – 1 non-leaves 5 21 2h leaves 2 9 16 25 1 3 7 10 13 19 22 30 40 Heap Structure Property • A binary heap is a complete binary tree. Complete binary tree – binary tree that is completely filled, with the possible exception of the bottom level, which is filled left to right. Examples: 41 Representing Complete Binary Trees in an Array 1 A From node i: 2 B 3 C 6 7 left child: 4 D 5 E F G 8 9 10 11 12 right child: H I J K L parent: implicit (array) implementation: A B C D E F G H I J K L 0 1 2 3 4 5 6 7 8 9 10 11 12 13 42 Why this approach to storage? 43 Heap Order Property Heap order property: For every non-root node X, the value in the parent of X is less than (or equal to) the value in X. 10 10 20 80 20 80 40 60 85 99 30 15 50 700 not a heap 44 Heap Operations • findMin: • insert(val): percolate up. • deleteMin: percolate down. 10 20 80 40 60 85 99 50 700 65 45 Heap – Insert(val) Basic Idea: 1. Put val at “next” leaf position 2. Percolate up by repeatedly exchanging node until no longer needed 46 Insert: percolate up 10 20 80 40 60 85 99 50 700 65 15 10 15 80 40 20 85 99 50 700 65 60 47 Insert Code (optimized) void insert(Object o) { int percolateUp(int hole, assert(!isFull()); Object val) { while (hole > 1 && size++; val < Heap[hole/2]) newPos = Heap[hole] = Heap[hole/2]; percolateUp(size,o); hole /= 2; } Heap[newPos] = o; return hole; } } runtime: (Code in book) 48 Heap – Deletemin Basic Idea: 1. Remove root (that is always the min!) 2. Put “last” leaf node at root 3. Find smallest child of node 4. Swap node with its smallest child if needed. 5. Repeat steps 3 & 4 until no swaps needed. 49 DeleteMin: percolate down 10 20 15 40 60 85 99 50 700 65 15 20 65 40 60 85 99 50 700 50 DeleteMin Code (Optimized) Object deleteMin() { int percolateDown(int hole, assert(!isEmpty()); Object val) { while (2*hole <= size) { returnVal = Heap[1]; left = 2*hole; size--; right = left + 1; newPos = if (right ≤ size && Heap[right] < Heap[left]) percolateDown(1, target = right; Heap[size+1]); else Heap[newPos] = target = left; Heap[size + 1]; if (Heap[target] < val) { return returnVal; Heap[hole] = Heap[target]; } hole = target; } else runtime: break; } (code in book) return hole; 51 } Insert: 16, 32, 4, 69, 105, 43, 2 0 1 2 3 4 5 6 7 8 52 Data Structures Binary Heaps 53 Building a Heap 12 5 11 3 10 6 9 4 8 1 7 2 54 Building a Heap • Adding the items one at a time is O(n log n) in the worst case • I promised O(n) for today 55 Working on Heaps • What are the two properties of a heap? – Structure Property – Order Property • How do we work on heaps? – Fix the structure – Fix the order 56 BuildHeap: Floyd’s Method 12 5 11 3 10 6 9 4 8 1 7 2 Add elements arbitrarily to form a complete tree.
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