The document discusses binary trees and their properties. It defines binary trees and provides terminology used to describe them, such as nodes, levels, paths, and heights. It describes different types of binary trees like complete, full, and binary search trees. It discusses common traversal methods for binary trees like breadth-first and depth-first. It also covers insertion and deletion of nodes in binary trees, describing approaches like merging subtrees and copying values.

1. Binary TreesBinary Trees
Chapter 6Chapter 6

2. Linked Lists SuckLinked Lists Suck
 By now you realize that the title to this slide isBy now you realize that the title to this slide is
true…true…
 When we are talking about searching orWhen we are talking about searching or
representing data structures that need arepresenting data structures that need a
hierarchical structures.hierarchical structures.
 We need a better structure…We need a better structure…
 So we get binary treesSo we get binary trees

3. Tree definitionTree definition
 Here is a (recursive, of course) definition forHere is a (recursive, of course) definition for
a tree:a tree:
1.1. An empty structure is an empty treeAn empty structure is an empty tree
2.2. If t1,…,tk are disjointed trees, then theIf t1,…,tk are disjointed trees, then the
structure whose root has as its children thestructure whose root has as its children the
roots of t1,…,tk is also a treeroots of t1,…,tk is also a tree
3.3. Only structures generate by rules 1 and 2 areOnly structures generate by rules 1 and 2 are
trees.trees.

4. More terminologyMore terminology
 Each node has to be reachable from the rootsEach node has to be reachable from the roots
through a unique sequence of arcs called athrough a unique sequence of arcs called a
path.path.
 The number of arcs in a path is called theThe number of arcs in a path is called the
length of the path.length of the path.
 The level of a node is the length of the pathThe level of a node is the length of the path
from the root to the node plus 1.from the root to the node plus 1.
 The height of a non-empty tree is theThe height of a non-empty tree is the
maximum level of a node in the tree.maximum level of a node in the tree.

5. Special TreesSpecial Trees
 An empty tree has a height of zero.An empty tree has a height of zero.
 A single node tree is a tree of height 1.A single node tree is a tree of height 1.
 This is the only case where a node is both a rootThis is the only case where a node is both a root
and a leaf.and a leaf.

6. Binary TreesBinary Trees
 According to the definition of trees, a node canAccording to the definition of trees, a node can
have any number of children.have any number of children.
 A binary tree is restricted to only having 0, 1,A binary tree is restricted to only having 0, 1,
or 2 children.or 2 children.
 A complete binary tree is one where all theA complete binary tree is one where all the
levels are full with exception to the last levellevels are full with exception to the last level
and it is filled from left to right.and it is filled from left to right.
 A full binary tree is one where if a node has aA full binary tree is one where if a node has a
child, then it has two children.child, then it has two children.

7. Full Binary Tree TheoremFull Binary Tree Theorem
 For all the nonempty binary trees whoseFor all the nonempty binary trees whose
nonterminal node have exactly two nonemptynonterminal node have exactly two nonempty
children, the number of leaveschildren, the number of leaves mm is greateris greater
than the number of nonterminal nodethan the number of nonterminal node kk andand mm
== kk + 1.+ 1.

8. Binary Search TreesBinary Search Trees
 A binary search tree (BST) is a binary tree thatA binary search tree (BST) is a binary tree that
has the following property: For each nodehas the following property: For each node nn ofof
the tree, all values stored in its left subtree arethe tree, all values stored in its left subtree are
less than valueless than value vv stored instored in nn, and all values, and all values
stored in the right subtree are greater thanstored in the right subtree are greater than vv..
 This definition excludes the case of duplicates.This definition excludes the case of duplicates.
They can be include and would be put in theThey can be include and would be put in the
right subtree.right subtree.

9. Binary Tree TraversalsBinary Tree Traversals
 A traversal is where each node in a tree isA traversal is where each node in a tree is
visited and visited oncevisited and visited once
 For a tree of n nodes there are n! traversalsFor a tree of n nodes there are n! traversals
 Of course most of those are hard to programOf course most of those are hard to program
 There are two very common traversalsThere are two very common traversals
 Breadth FirstBreadth First
 Depth FirstDepth First

10. Breadth FirstBreadth First
 In a breadth first traversal all of the nodes on aIn a breadth first traversal all of the nodes on a
given level are visited and then all of the nodesgiven level are visited and then all of the nodes
on the next level are visited.on the next level are visited.
 Usually in a left to right fashionUsually in a left to right fashion
 This is implemented with a queueThis is implemented with a queue

11. Depth FirstDepth First
 In a depth first traversal all the nodes on aIn a depth first traversal all the nodes on a
branch are visited before any others are visitedbranch are visited before any others are visited
 There are three common depth first traversalsThere are three common depth first traversals
 InorderInorder
 PreorderPreorder
 PostorderPostorder
 Each type has its use and specific applicationEach type has its use and specific application

12. InsertionInsertion
 In order to build a tree you must be able toIn order to build a tree you must be able to
insert into the treeinsert into the tree
 In order to do this you need to know where theIn order to do this you need to know where the
nodes goesnodes goes
 Typically the tree is searched looking for aTypically the tree is searched looking for a
null pointer to hang the new element fromnull pointer to hang the new element from
 There are two common ways to do thisThere are two common ways to do this
 Use a look ahead or check for null as the firstUse a look ahead or check for null as the first
line in the codeline in the code

13. More insertionMore insertion
 I prefer to check for null as the first thing I doI prefer to check for null as the first thing I do
in my codein my code
 It simplifies some of the testsIt simplifies some of the tests
 And makes for a really easy to check for baseAnd makes for a really easy to check for base
casecase

14. CodeCode
InsertionHelper( Node *n, T data )InsertionHelper( Node *n, T data )
{{
if ( node == 0 )if ( node == 0 )
return new Node( data );return new Node( data );
if ( n->getData() < data )if ( n->getData() < data )
setLeft( InsertionHelper( n->getLeft(), data);setLeft( InsertionHelper( n->getLeft(), data);
elseelse
setRight( InsertionHelper( n->getRight(), data);setRight( InsertionHelper( n->getRight(), data);
}}

15. DeletionDeletion
 Deletion poses a bigger problemDeletion poses a bigger problem
 When we delete we normally have two choicesWhen we delete we normally have two choices
 Deletion by mergingDeletion by merging
 Deletion by copyingDeletion by copying

16. Deletion by MergingDeletion by Merging
 Deletion by merging takes two subtrees andDeletion by merging takes two subtrees and
merges them together into one treemerges them together into one tree
 The idea is you have a node n to deleteThe idea is you have a node n to delete
 N can have two childrenN can have two children
 So you find the smallest element in n’s leftSo you find the smallest element in n’s left
subtreesubtree
 You then take n’s right subtree and merge it toYou then take n’s right subtree and merge it to
the bottom of the left subtreethe bottom of the left subtree
 The root of the left subtree replaces nThe root of the left subtree replaces n

17. Deletion by copyingDeletion by copying
 This will simply swap values and reduce aThis will simply swap values and reduce a
difficult case to an easier onedifficult case to an easier one
 If the node n to be deleted has no children,If the node n to be deleted has no children,
 easy blow it awayeasy blow it away
 If it has one childIf it has one child
 Easy simply pass n’s child pointer up, make n’sEasy simply pass n’s child pointer up, make n’s
parent point to n’s child and blow n awayparent point to n’s child and blow n away
 If n has two child,If n has two child,
 Now we have deletion by copyingNow we have deletion by copying

18. DetailsDetails
 We find the smallest value in n’s right subtreeWe find the smallest value in n’s right subtree
 We will take the value from that node and putWe will take the value from that node and put
it in place of the value in nit in place of the value in n
 We will then blow away the node that had theWe will then blow away the node that had the
smallest value in itsmallest value in it