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Data Structures Trees Basics

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Published Date:09-07-2017
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csci 210: Data Structures Trees 1Summary  Topics • general trees, definitions and properties • interface and implementation • tree traversal algorithms • depth and height • pre-order traversal • post-order traversal • binary trees • properties • interface • implementation • binary search trees • definition • h-n relationship • search, insert, delete • performance  READING: • GT textbook chapter 7 and 10.1 2Trees  So far we have seen linear structures • linear: before and after relationship • lists, vectors, arrays, stacks, queues, etc  Non-linear structure: trees • probably the most fundamental structure in computing • hierarchical structure • Terminology: from family trees (genealogy) 3Trees root  store elements hierarchically  the top element: root  except the root, each element has a parent  each element has 0 or more children 4Trees  Definition • A tree T is a set of nodes storing elements such that the nodes have a parent-child relationship that satisfies the following • if T is not empty, T has a special tree called the root that has no parent • each node v of T different than the root has a unique parent node w; each node with parent w is a child of w  Recursive definition • T is either empty • or consists of a node r (the root) and a possibly empty set of trees whose roots are the children of r  Terminology • siblings: two nodes that have the same parent are called siblings • internal nodes • nodes that have children • external nodes or leaves • nodes that don’t have children • ancestors • descendants 5Trees root internal nodes leaves 6Trees ancestors of u u 7Trees descendants of u u 8Application of trees  Applications of trees • class hierarchy in Java • file system • storing hierarchies in organizations 9Tree ADT  Whatever the implementation of a tree is, its interface is the following • root() • size() • isEmpty() • parent(v) • children(v) • isInternal(v) • isExternal(v) • isRoot() 10Tree Implementation class Tree TreeNode root; //tree ADT methods.. class TreeNodeType Type data; int size; TreeNode parent; TreeNode firstChild; TreeNode nextSibling; getParent(); getChild(); getNextSibling(); 11Algorithms on trees: Depth  Depth: • depth(T, v) is the number of ancestors of v, excluding v itself  Recursive formulation • if v == root, then depth(v) = 0 • else, depth(v) is 1 + depth (parent(v))  Compute the depth of a node v in tree T: int depth(T, v)  Algorithm: int depth(T,v) if T.isRoot(v) return 0 return 1 + depth(T, T.parent(v))  Analysis: • O(number of ancestors) = O(depth_v) • in the worst case the path is a linked-list and v is the leaf • == O(n), where n is the number of nodes in the tree 12Algorithms on trees: Height  Height: • height of a node v in T is the length of the longest path from v to any leaf  Recursive formulation: • if v is leaf, then its height is 0 • else height(v) = 1 + maximum height of a child of v  Definition: the height of a tree is the height of its root  Compute the height of tree T: int height(T,v)  Height and depth are “symmetrical”  Proposition: the height of a tree T is the maximum depth of one of its leaves. 13Height  Algorithm: int height(T,v) if T.isExternal(v) return 0; int h = 0; for each child w of v in T do h = max(h, height(T, w)) return h+1;  Analysis: • total time: the sum of times spent at all nodes in all recursive calls • the recursion: • v calls height(w) recursively on all children w of v • height() will eventually be called on every descendant of v • overall: height() is called on each node precisely once, because each node has one parent • aside from recursion • for each node v: go through all children of v – O(1 + c_v) where c_v is the number of children of v • over all nodes: O(n) + SUM (c_v) – each node is child of only one node, so its processed once as a child – SUM(c_v) = n - 1 14 • total: O(n), where n is the number of nodes in the treeTree traversals  A traversal is a systematic way to visit all nodes of T.  pre-order: root, children • parent comes before children; overall root first  post-order: children, root • parent comes after children; overall root last void preorder(T, v) visit v for each child w of v in T do preorder(w) void postorder(T, v) for each child w of v in T do postorder(w) visit v Analysis: O(n) same arguments as before 15Examples  Tree associated with a document Pape r Title Abstract Ch1 Ch2 Ch3 Refs 1.1 1.2 3.1 3.2  In what order do you read the document? 16Example  Tree associated with an arithmetical expression + 3 - + 12 5 1 7  Write method that evaluates the expression. In what order do you traverse the tree? 17Binary trees 18Binary trees  Definition: A binary tree is a tree such that • every node has at most 2 children • each node is labeled as being either a left chilld or a right child  Recursive definition: • a binary tree is empty; • or it consists of • a node (the root) that stores an element • a binary tree, called the left subtree of T • a binary tree, called the right subtree of T  Binary tree interface • left(v) • right(v) • hasLeft(v) • hasRight(v) • + isInternal(v), is External(v), isRoot(v), size(), isEmpty() 19Properties of binary trees  In a binary tree • level 0 has = 1 node d=0 • level 1 has = 2 nodes • level 2 has = 4 nodes d=1 • ... • level i has = 2i nodes d=2 d=3  Proposition: Let T be a binary tree with n nodes and height h. Then h+1 • h+1 = n = 2 -1 • lg(n+1) - 1 = h = n-1 20