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

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1 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 asy access (similar to binary search in 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 ncestor 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 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) 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 vis an internal node 5 6 Tree bstract ata Type Linked Structure for Trees Generic methods (not necessarily related to a tree structure): method description ismpty() Test whether the tree has any nodes or not size() Return the number of nodes in the tree iterator() Return an iterator of all the elements stored in the tree positions() Return an iterable collection of all the nodes of the tree replace(v,e) Replace with eand return the element stored at node v node is represented by an object storing lement Parent node Sequence of children nodes (element) (parent node) (a list of children) (child node) 7 8

3 Linked Structure for Trees inary Tree efinition a tree in which each node has at most two children ach child is either the left child or the right child of its parent. F F left child right child 9 10 rithmetic xpression Tree ecision Tree inary tree associated with an arithmetic expression Internal nodes: operators xternal nodes: operands xample: arithmetic expression tree for the expression: (2 (a 1) (3 b)) + inary tree associated with a decision process Internal nodes: questions with yes/no answer xternal nodes: decisions xample: dining decision On a diet? Yes Want a fast meal? No On expense account? 2 3 b Yes No Yes No a 1 Subway Mconald s????????????????? 11 12

4 Proper inary Trees ach internal node has exactly 2 children. Properties e i 1 n 2e 1 h i h (n 1)2 e 2 h h log 2 e h log 2 (n 1) 1 n :number of total nodes e :number of external nodes i :number of internal nodes h :height (maximum depth of a node) Linked Structure for inary Trees node is represented by an object storing lement Parent node Left child node Right child node Left child Parent node lement Right child Linked Structure for inary Trees inary Tree Implementation public class Node { int value; Node left; Node right; // data stored at node // pointer to left child // pointer to right child } public Node(int i) { value = i; left = null; right = null; } 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; } 15 16

5 public class inarytree { Node root; } inary Tree Implementation public Tree() { 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) {... } n rray ased Representation Nodes are stored in an array. Node v is stored at [rank(v)] rank(root) = 1 Left in even: if node is the left child of parent(node), rank(node) = 2 rank(parent(node)) Right in odd: if node is the right child of parent(node), rank(node) = 2 rank(parent(node)) 1 [0] is always empty [i] is empty if there is no node in the ith position The array size N is 2 (h+1) n rray ased Representation Tree alance Full binary tree (perfect binary tree) binary tree of height h is full if every node has exactly two children and all leaf nodes have the same level. 4 F J G H G H

6 omplete binary tree Tree alance binary tree is complete if it is full to level h 1 and level h is filled fro m the left with contiguous nodes

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

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