Binary Trees
Linked list are unary trees!
Corresponding Book
Sections
• Data Structures: Section 7.1, 7.2
What Are Trees?
A tree is a hierarchical data structure made
up of nodes with parent and child
relationships/references to other nodes
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Tree Node Vocabulary
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Parent node - the node a node descends from in the tree
Child node - node that descends from a given node in the tree
Root node - node at the top of the tree
Internal node - node with at least one child
External node (leaf) - node with no children
Sibling nodes - nodes with the same parent
Ancestor - parent, grandparent, etc. of a node
Descendent - child, grandchild, etc. of a node
Tree Vocabulary
• Height - How how many nodes away from a leaf
node that a node is
• Depth - How many nodes away from the root node
that a node is
• Subtree - Tree formed by taking a node from a
given tree, as well as all descendants of that node
Analyzing a Tree
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What is the root node?
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What are the children of
node 3? Descendants?
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Who is the parent of
node 6? Ancestors?
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What is the height of this
tree? What is the depth
of node 8?
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What are the siblings of
node 8?
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Examples of Tree Usage in
Computer Science
• File Systems - B-Trees
• Java TreeMap - Red-Black Tree
• Machine Learning - Decision Trees
• AI - Mini-Max Trees
• Heaps - We will get to this later in the semester
• Printing tests!
Decision Tree
Temperature
<35F
Wind
Direction
Temperature
<20F
Wind
Speed
<4
Walk
Drive
>=20F
East
Precipitation
>=4
No
Yes
>=3
Walk
Not East
Wind
Speed
Precipitation
No
Wind
Speed
Drive
>=35F
Walk
<3
Drive
Yes
Drive
<6
Walk
>=6
Drive
Trie (Dictionary Tree)
http://blog.ivank.net/trie-in-as3.html
Tree Traversals
• Pre-order - A node is visited before its
descendants
• Post-order - A node is visited after its
descendants
Tree Traversals
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Assuming left to right child traversal, what is the Pre Order traversal?
What is the Post Order Traversal?
Binary Search Tree
• A Binary Tree is where a node has up to
two children only
• A Binary Search Tree is a tree where it is
maintained that every value to the left
(down the left child) of a node is less than
the node, and every value to the right (down
the right child of node is greater than the
node
Binary Search Tree
Example
D
B
A
E
C
F
Binary Search Tree
Example
E
D
C
B
A
Big Oh of Binary
Search Trees
• Lookup time
• O(lg n) if balanced
• O(n) possible worst case if not balanced
• Insertion time - same thing!
• Deletion time? Will revisit after coding!
In Order Traversal
• Nodes are visited according to their logical
ordering
• How would you write the in-order traversal
of a Binary Search Tree?
• What should be printed first? second? last?
• Write the pseudo-code!