Data Structures and Algorithmsgarryowen.csis.ul.ie/~cs4115/resources/lect10.pdf · Announcements Abstract Data Types (ADTs) Outline 1 Announcements 2 Abstract Data Types (ADTs) Trees

AnnouncementsAbstract Data Types (ADTs)

Data Structures and Algorithms

P. Healy

CS1-08Computer Science Bldg.

tel: [email protected]

Spring 2018–2019

P. Healy (University of Limerick) CS4115 Spring 2018–2019 1 / 21


Outline

1 Announcements

2 Abstract Data Types (ADTs)Trees

Implementing TreesTree Traversals

Binary TreesBinary Search Trees



Announcements

Mid-term: bubbling up to the surface of my head



TreesBinary TreesBinary Search Trees

Outline

1 Announcements







Introduction

A tree is defined recursively as follows:A tree is a (possibly empty) collection ofIf non-empty, the tree has one special node, r , the root, andZero or more subtrees, T1,T2, . . . ,TkSubtrees T1, . . . ,Tk are connected to r by a directed edge(from r to Ti )

The root node of a subtree is a child of r and r is theparent of each subtree root

T1 T2 T3 T4

r




Introduction (contd.)

A tree of n nodes has edges since each nodeexcept r has a parent node (denoted by an edge)Nodes may have an arbitrary number of childrenNodes with are called leaf nodesLeaf nodes on 13-node tree over are E, F, K, H, I, L and MNodes with the same parent are called siblingsA path, p, from node n1 to node nk is the sequence ofnodes n1,n2, . . . ,nk such that ni is the parent of ni+1,where 1 ≤ i < kThe length of p is k − 1The length of the path from a node to itself is 0




Introduction (contd.)A

B C D

E F G H I J

K L M

d=0

d=1

d=2

d=3

There is exactly one path from r to every nodeThe depth of a node, n, is the length of the path from r to nThe height of n is the length of longest path from n to a(descendant) leafNode D (above) is at depth 1 and height 2If there is a path from n1 to n2 then n1 is a (proper)ancestor of n2 and n2 is a (proper) descendant of n1 (ifn1 6= n2)




A Possible Tree Implementation

Could think of earlier tree as followsA

C D

F G H I J

L M

B

E

K

This suggests the following structstruct TreeNode // unwieldy implementation{Object el;TreeNode* firstChild;TreeNode* nextSibling;

}No need for anything more than two pointers




A Better Tree Implementation

The previous struct1 could have been useful inpre-C++ daysThe struct below requires a class but is morenatural way of thinking

struct TreeNode // more natural{

Object el; // the datalist<TreeNode> children;

}

1A struct can be thought of as a C++ object without member functionsP. Healy (University of Limerick) CS4115 Spring 2018–2019 9 / 21



Systematically Processing a Tree

Since a tree is defined recursively, to traverse a tree wecan traverse each of its subtrees using the same algorithmThis traverses the tree depthwiseWe can “process” or “visit” the root node of each (sub)treeeither before or after we traverse its childrenProcessing the root node before (after, respectively)traversing the children is called a preorder (post-order)To “describe” or print out the names of the nodes of a treeusing preorder traversal, the code on the following slide isa rough cut at itWhat’s important is the position of the recursive call:colours match up with above




Systematically Processing a Tree (contd.)

On next slide we see pseudo-C++ code for traversing atree that is represented using the second (more natural)TreeNode struct on earlier slideGiven the root of a tree we iterate over its childrentraversing each subtreeEither before that or after we print out the root’s name,indented by depth spacesObserve difference in outputs (two slides on)

Rabhadh / Attention / Achtung / Warning: Do not attempt tocompile this code





void sideways_draw_tree(const TreeNode& root,const int depth = 0)

{// use depth as indentation!!indented_print_name(root.el, depth);// cout« namefor (root.children().first(); // get first child

!root.children(); // more children?++root.children()) // get next child

sideways_draw_tree(root.children(), depth+1);// indented_print_name(root.el, depth);// cout« name}





sideways_draw_tree() traversal of previous 13-node treepre-order

|A| B| E| C| F| G| K| D| H| I| J| L| M

post-order (not just backwards pre-order!)

| E| B| F| K| G| C| H| I| L| M| J| D|A




Outline

1 Announcements







When We Limit the Number of Children...

In a binary tree a node can have at most two childrenThese are referred to as the nodes left and right childrenOn i th level of a binary tree can have at most nodesOver entire tree, number of nodes on k levels, N(k), is

N(k) =k−1∑i=0

2i = 2k − 1

So, 2k − 1 nodes can be stored using k levels

n ≤ 2k − 1

What about the other way? How many levels do we needto store n nodes?




When We Limit the Number of Children... (contd.)

From above, n ≤ 2k − 1⇒ 2k ≥ n + 1⇒ k ≥ log(n + 1)So putting 7 nodes in a binary tree cannot be done withfewer than log2 8 = 3 levels; putting 8 nodes in a treerequires minimum of log2 9 = 3.1699→ 4 levels!...must round up if necessary to next highest integer

Need at least

k ≥ dlog(n + 1)e

levels to store n nodesTherefore the to the node in a binarytree can be O(log n)This makes a binary tree potentially a useful candidate forstoring information

7 Common mistake: an n-node binary tree does not havedepth O(log n) in general

3 We want it to have depth O(log n) but we will need to dosome work to achieve it




Binary Tree Examples

A

C D

F G H I J

L M

B

E

K

Tree seen earlier: n-ary treescan be faked to use just twopointers; but now siblings (e.g.,B and C) have different depth inbinary representation

null

null

null

null null

A binary tree with height, h = n




Outline

1 Announcements







A Third Type of Tree Traversal

With a binary tree we have a type of traversal:inorderAn inorder traversal of a tree recursively traverses the leftsubtree, “visits” or “processes” the tree root and traversesthe right subtreeA binary tree is a binary search tree if an inorder traversalof the tree visits the nodes in sorted orderImportant observation: if inthe sorted order then either

the location of x is the rightmost node in y ’s left subtree, orthe location of y is the leftmost node in x ’s right subtree




A Third Type of Tree Traversal (contd.)

D

G

N

M

B

H

D

G

N

M

B

H

Binary Search Tree Binary Tree

Traversals of left and right trees above (assuming that weprocess siblings left to right):

left rightPreorder MDBGHN MDBGHNInorder BDGHMN BDHGMNPostorder BHGDNM BHGDNM




A bad and good BST

Abigail

Adam

Agnes

Alex

Alice

Angela

Arthur

Alex

Adam

Abigail Agnes

Angela

Alice Arthur

Tree has k = dlog(n + 1)e = 3levels, the smallest possible;this shows us that with luck(??)we can achieve a very denseand shallow tree.


Documents

Data Structures and Algorithmsgarryowen.csis.ul.ie/~cs4115/resources/lect10.pdf · Announcements Abstract Data Types (ADTs) Outline 1 Announcements 2 Abstract Data Types (ADTs) Trees