A Hierarchical Structure. Lecture11: Tree I. Tree Data Structures. Unix/Linux file systems. Bohyung Han CSE, POSTECH

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1 Unix/Linux file systems Hierarchical Structure (2015F) Lecture11: Tree I ohyung Han S, POSTH bhhan@postech.ac.kr 2 Tree ata Structures n abstract model of a hierarchical structure 2 dimensional structure asy access (similar to array) asy insert and removal (similar to linked list) Tree ata Structures n abstract model of a hierarchical structure Trees consist of nodes and links denoting parent child relation Special case of graphs without loops ach element (except root) has a parent and zero or more children. omputers R Us omputers R Us Sales Manufacturing R& Sales Manufacturing R& US International Laptops esktops US International Laptops esktops urope sia anada urope sia anada 3 4

2 Terminology Root: node without parent Internal node: node with at least one child,,,f Leaf: node without children,i,j,k,g,h, Subtree: tree consisting of a node and its descendants ncestors of a node: parent, grandparent, grand grandparent, escendant of a node: child, grandchild, grand grandchild, epth (level) of a node: number of ancestors Height of a tree: maximum depth of any node 3 F I J K G Subtree H ata structure Tree bstract ata Type Node: data element and link to children Root: a special node List of methods method root() parent(v) children(v) isroot(v) isxternal(v) isinternal(v) ismpty() size() description Return the tree s root; error if tree is empty Return v s parent; error if vis a root Return v s children (an iterable collection of nodes) Test whether vis a root Test whether vis an external node Test whether v is an internal node Test whether the tree has any nodes or not Return the number of nodes in the tree 5 6 Linked Structure for Trees Linked Structure for Trees node is represented by an object storing lement Parent node Sequence of children nodes Parent node F F list of children lement child node 7 8

3 efinition inary Tree tree in which each node has at most two children ach child is either the left child or the right child of its parent. rithmetic xpression Tree inary tree associated with an arithmetic expression Internal nodes: operators xternal nodes: operands xample: arithmetic expression tree for the expression: Left child Right child 2 3 b a ecision Tree inary tree associated with a decision process Internal nodes: questions with yes/no answer xternal nodes: decisions xample: dining decision Linked Structure for inary Trees node is represented by an object storing lement Parent node Left child node Right child node Parent node Want a fast meal? Yes No On a diet? On expense account? Yes No Yes No Subway Mconald s???????????????? Left child lement Right child 11 12

4 Linked Structure for inary Trees inary Tree Implementation public class Node { int value; Node left; Node right; public Node(int i) { value = i; left = null; right = null; } // data stored at node // pointer to left child // pointer to right child } public Node getlefthild(node node) { return left; } public Node getrighthild(node node) { return right; } public void setlefthild(node node) { left = node; } public void setrighthild(node node) { right = node; } public class inarytree { Node root; } inary Tree Implementation public inarytree() { root = null; } // insert an element to the current tree public void insert(int i) {... } // deleted a node from the current tree public Node delete(int i) {... } // find a node containing a particular value public Node find(int i) {... } asic idea n rray ased Representation ach node has up to 2 children and can be mapped with a particular location in an array. Implementation Nodes are stored in an array. Node is stored at rank. rank root 1 Left in even: if a node is the left child of a parent node, rank node 2 rankparent node Right in odd: if node is the right child of a parent node, rank node 2 rank parent node 1 0 is always empty is empty if there is no node in the th position 15 16

5 n rray ased Representation F J Perfect binary tree Tree alance binary tree of height h is perfect if every node has exactly two children and all leaf nodes have the same level. Perfect binary tree is sometimes called full binary tree G H F J G H Tree alance Tree alance omplete binary tree binary tree is complete if it is full to level h 1 and level h is filled from the left with contiguous nodes. The full binary tree is a special case of complete binary tree. Proper binary tree binary tree in which each node has exactly zero or two children The full binary tree is a special case of proper binary tree

6 Proper inary Trees ach internal node has exactly 2 children. Properties /2 2 log log 1 1 : number of total nodes : number of external nodes : number of internal nodes : height (maximum depth of a node) 21 22

A Hierarchical Structure. Lecture11: Trees I. Tree Data Structures. Unix/Linux file systems. Bohyung Han CSE, POSTECH

A Hierarchical Structure. Lecture11: Trees I. Tree Data Structures. Unix/Linux file systems. Bohyung Han CSE, POSTECH Unix/Linux file systems Hierarchical Structure (2013F) Lecture11: Trees I ohyung Han S, POSTH bhhan@postech.ac.kr 2 Tree ata Structures n abstract model of a hierarchical structure 2 dimensional structure