Data Structures – Binary Tree
What is a tree?
Where do you see trees?
• Ummm...outside
• file systems
Tree Terminology
• node – any element of the tree
• root – the topmost node of the tree
• parent – a node that has one or more nodes
connected below it (children)
• child – a node that has a connected node above
it (parent)
• leaf – any child node at the bottom of the tree
• subtree – a parent and all the nodes below it
Name that part!
root /
parent
subtree
parent /
child’
child /
leaf
child /
leaf
child /
leaf
So what’s a binary tree?
1. A parent can have at most TWO children (left
child, right child)
2. The left child’s data and all the data in the left
subtree is “less than” the parent’s data
3. The right child’s data all the data in the right
subtree is “greater than” the parent’s data
• NOTE: The “less than” and “greater than”
requirements of the subtrees is commonly
known outside of the IB world as a Binary
Search Tree
Example – Valid Binary Tree
6
2
1
10
4
Example – INVALID Binary Tree
3
2
1
10
4
Practice 1
• Create a binary tree by inserting the
following numbers in the given order:
6, 4, 2, 7, 8, 14, 9
Practice 2
• Create a binary tree by inserting the
following numbers in the given order :
14, 75, 2, 34, 25, 26, 27, 28
Another Binary Tree
h
d
a
m
k
x
Practice 3
• Create a binary tree by inserting the
following strings:
Wanda, Alpos, Vazbyte, Fecso, Downs,
Mata, Montante
Practice 4
• Create a binary tree by inserting the
following strings:
Ramos, Dia, Khurelbaatar, Nice,
Ren, Shahid, Zetkulic
Adding to a Binary Tree
1. Start at the root
2. Compare against the current node
1. Go left if less than current node OR insert if
there is none (creating a new leaf)
2. Go right if greater than current node OR
insert if there is none (creating a new leaf)
3. Repeat step 2
Searching in a Binary Tree
1. Start at the root
2. Compare against the current node
1. Found the node if they are equal
2. Go left if less than current node OR if there is
no left, then does not exist
3. Go right if greater than current node OR if
there is no left, then does not exist
3. Repeat step 2
Search(4) – What path do you take?
6
2
1
10
4
13
Search(9) – What path do you take?
6
2
1
10
4
13
Removing from a Binary Tree
1. Search for matching node
2. If node is a leaf, then unlink its parent
3. If node is a parent of one child, then link
the node’s parent to the node’s child
4. If node is a parent of two children, then
travel down its right subtree to find the
left-most leaf (smallest value of the right
subtree). Take the value and put it in the
original node that was being removed.
Unlink the right-most leaf that you found.
Remove(4) – 0 children case
6
2
1
10
4
13
Remove(10) – 1 child case
6
2
1
10
4
13
Remove(6) – 2 children case
6
2
1
10
4
13
When do we use binary trees?
• ...whenever we need to search for quickly
– Inherent binary searching capabilities
– Large trees can be searched quickly
• What is the best case scenario?
• What is the worst case scenario?
• What is the average case scenario?
• Compare a Linked List to a Binary Tree
Tree Traversal
• Add, remove, search only go down 1 path
• How do you “walk-through” all the nodes
of a tree?
In-order tree traversal
1. Left-subtree traversal if it exists and rerun
2. Action on the current node (e.g. print)
3. Right-subtree traversal if it exists and
rerun
Note: In-order traversal often used to visit
nodes in their inherent order
In-order Traversal (1, 2, 4, 6, 8, 10, 13)
6
2
1
10
4
8
13
Pre-order tree traversal
1. Action on the current node (e.g. copy)
2. Left-subtree traversal if it exists and rerun
3. Right-subtree traversal if it exists and
rerun
Note: Pre-order traversal is often used to
duplicate a tree
Pre-order Traversal (6, 2, 1, 4, 10, 8, 13)
6
2
1
10
4
8
13
Post-order tree traversal
1. Left-subtree traversal if it exists and rerun
2. Right-subtree traversal if it exists and
rerun
3. Action on the current node (e.g. print)
Note: Post-order traversal is often used to
completely delete or free up all nodes by
visiting children and lowest levels first.
(not really necessary with garbage
collection)
Post-order Traversal (1, 4, 2, 8, 13, 10, 6)
6
2
1
10
4
8
13
Other Resources
• http://www.csanimated.com/animation.php
?t=Binary_search_tree