Joe Meehean
1
A
B
D
I
C
E
X
 Terminology
• each circle isAa node
• pointers are edges
• topmost node is the root
• bottom
B nodesI are leavesC
 no outgoing edges
 Every
non-empty tree has
• one root
D
E
• one or more leaves
X
 Node
A is the parent
of node B
 Node B is the child of
node A
 The root has no
parent
 All other nodes have
exactly 1 parent
A
B
 Not
a tree
 D has 2 parents
A
B
C
D
 Path
is a sequence of
connected nodes
 Length of a path is
the number of edges
in the path
• A to D is 2
• C to F is 1
Tree 1
Tree 2
A
C
B
F
Tree 3
• X is 0
D
X
 Height
of a tree:
length of its longest
path from root to leaf
 Ex: Height of 2
A
B
E
C
D
 Depth
of a node:
length of path from
root
 Example
A
B
• A: 0
• B: 1
C
• E: 2
• D: 2
• C: 1
E
D
 Subtrees
of a node
are the nodes rooted
at a nodes children
 As subtrees
A
B
• rooted at B
• rooted at C
E
C
D
 Special Tree
A
 No
node has more
than 2 children
• can have 0,1, or 2
 Each
child is either
“right” or “left”
B
E
C
D
A
B
E
X
C
D
Y
Z
S
T
template <typename D>
class BinaryTreenode<D>{
private:
D data_;
BinaryTreenode<D>* left_;
BinaryTreenode<D>* right_;
public:
BinaryTreenode(D d = D(),
BinaryTreenode<D>* left = NULL,
BinaryTreenode<D>* right = NULL);
};
 Iterate
through all nodes
• each node visited once, to…
• print all values
• see if a node has some property
• modify the node, etc…
4
common orders for visiting nodes
• preorder
• postorder
• in order (binary trees only)
• level order
 Depth-first
traversal
 Visit the root first
 Recursive definition
1
• visit the root
2
• do a preorder
A
B
C
5
traversal of each
subtree, left to right
 Ex: A
BEDCF
E
3
F
D
4
6
 Depth-first
traversal
 Visit the root last
 Recursive definition
6
• do a postorder
3
traversal of each
subtree, left to right
• visit the root
 Ex: E
DBFCA
B
E
1
A
F
D
2
C
5
4
 Depth-first
traversal
 For binary trees only
 Visit root in
between subtrees
 Recursive definition
4
2
• in-order traversal of
left subtree
• visit the root
• in-order traversal of
right subtree
 Ex: E
BDAFC
B
E
1
A
F
D
3
C
6
5
 Visit
all nodes at level 1 (depth 1)
• then level 2, level 3, etc…
• always left to right (or some order)
 Use
a queue
• instead of recursion (implicitly uses a stack)
q.push(root);
while( !q.empty() ){
//dequeue node n
//visit n
//enqueue all of n’s children(L to R)
}
 Special
kind of binary tree
 Each node stores a key
• sometimes an associated value
 For
each node n
• all keys in n’s left subtree are < key at n
• all keys in n’s right subtree are > key at n
• if duplicate keys allowed
 keys that equal n can go left XOR right (not both)
18
 Insert
a key (and associated data)
 Lookup a key (and associated data)
 Remove a key (…)
 Print all keys in sorted
• using an inorder traversal
19
4
2
1
6
5
4
2
6
9
5
4
2
9
7
3
Yes
In order traversal produces:
24569
No: 7 is not < 6
20
6
lookup(2)
4
3
9
5
7
15
2
21
6
lookup(2)
4
3
2
9
5
7
15
Eliminates half the nodes
at every compare
22
6
lookup(2)
4
3
2
9
5
7
15
Eliminates half the nodes
at every compare
23
6
lookup(2)
4
3
2
9
5
7
15
Eliminates half the nodes
at every compare
24
 Worst-case
• O(height of tree)
 Worst
of worst
• height is N
• lookup is O(N)
 Best
worst-case
• height is log2N
• lookup is O(logN)
 O(LogN)
is waaaay better than O(N)
25
 New
values inserted as leaves
 Must choose position to respect
BST ordering
• and to ensure we can find it with a lookup
 Duplicate
keys are not allowed
26
 Traverse
the tree
• like a lookup
 If
we find a duplicate
• return an error
 If
we end up at a null (child of a leaf)
• make a new node with the key
• make it the child of the leaf
 Note
the above two were our base cases
for lookup too
27
 Similar
to lookup
• worst-case follow path from root to leaf
• O(logN) for a balanced tree
• O(N) for a completely unbalanced tree
28
 Find
the node n w/ key to be deleted
 Different actions depending on
n’s # of kids
• Case 1: n has 0 kids (it’s a leaf)
 set parent’s n-pointer (left or right) to null
• Case 2: n has 1 kid
 set parent’s n-pointer to point to n’s only kid
• Case 3: n has 2 kids
 replace n’s key with a key further down in the tree
 delete that node
29
8
delete(17)
15
…
…
20
…
18
…
16
17
30
8
delete(17)
15
…
…
20
…
18
…
16
17
31
8
delete(16)
15
…
…
20
…
18
…
16
17
32
8
delete(16)
15
…
…
20
…
18
…
16
17
33
delete(15)
8
15
…
…
20
…
18
…
16
Smallest value in right subtree
17
34
delete(15)
8
16
…
…
20
…
18
…
16
Case 2: 1 kid
Replace 16 with it’s only child
17
35
8
delete(15)
16
…
…
20
…
18
…
16
Case 2: 1 kid
Replace 16 with it’s only child
17
36
 Worst-case
(delete root)
 Root
to leaf
 H = Height of Tree
• O(H)
 Balanced
tree
• O(logN) where N is keys in tree
 O(N)
for a completely unbalanced tree
37
 DSW
algorithm
• transform BST into a linked list
• transform linked list into balanced BST
• uses tree rotations
 Tree
rotation
• rotates a part of the tree
• makes a child the parent
• makes a parent the child
• preserves BST property
38
 Left-child
tree rotation: (4 & 6)
6
4
2
9
5
39
 Left-child
tree rotation: (4 & 6)
6
4
4
9
2
6
Make 4 parent of 6
2
5
5
9
40
 Left-child
tree rotation: (4 & 6)
6
4
2
4
9
5
Assign 4’s abandoned
child to 6.
2
5
6
9
41
 General
left-child tree rotation
A
k2
k1
B
X
Z
Y
42
 General
left-child tree rotation
A
k2
k2
k1
k1
B
X
B
Z
Y
X
A
Y
Z
43
 General
left-child tree rotation
A
k2
B
k2
k1
k1
B
X
Z
Y
A
X
Y
Z
44
 General
left-child tree rotation
A
k2
B
k2
k1
k1
B
X
Z
Y
A
X
Y
Z
45
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