1. Trees

2. What is a Tree?
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3. What is a Tree? Trees are an important data structure
They organize data hierarchically
They are used to
model hierarchical data in the real world
store and retrieve data
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4. A hierarchical organization of a hypothetical company
Computers”R”Us
Sales
R&D
Manufacturing
Laptops
Desktops
Local
International
Europe
Asia
Africa
A hierarchical organization of a hypothetical company
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5. Tree Terminology Node Child Parent Data stored in the tree
A node descended from another node
Parent
The node from which another node descended a c b
a is the parent of both b & c
b & c are children of a
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6. Tree Terminology Root The node from which all others descend Branch
Joins nodes
Leaf (External node )
A node without children
Degree=3
Maximum # of children
Siblings
Children of a common parent
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7. Tree Terminology Ancestors of a node Descendant of a node
Any node from which a node descended (parent, grandparent, grand-grandparent, etc).
Descendant of a node
Any child, grandchild, grand-grandchild, etc.
Depth of a node
Number of ancestors
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8. Tree Terminology Height of a tree Subtree Path
Maximum depth of any node
Subtree
Tree consisting of a node and its descendants
Path
The set of branches connecting an ancestor to a descendant
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9. Tree Terminology
Depth = 0
Depth = 2
Depth = 3
Subtree
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10. Trees vs. Graphs
A tree
Cannot contain cycles
Each node has 1 parent or it is at the root
Any structure which does not meet these criteria is a graph
The structures to the right are graphs
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11. Review: Recursive definition of list
Empty list (0 nodes) OR
Head node and 0 or 1 non-empty sublist
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12. Recursive definition of Tree
Empty tree (0 nodes) OR
Root node and 0 or more non-empty subtrees (child node and its descendents)
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13. Tree Traversal
To traverse any data structure means to visit every node in turn in a systematic manner
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14. Review: List Traversal
List – process sequentially by
iteration
2. recursion
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15. Tree Traversal
Pre-order
In-order
Post-order
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16. Pre-order Traversal Recursive definition: Visit root before children1 2 5 9 3 6 11 8 10 7 4
Recursive definition: Visit root before children
Algorithm preOrder(v)
visit(v)
for each child w of v
preOrder (w)
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17. Post-order Traversal Recursive definition: Visit root after children11 2 10 5 1 4 9 6 7 8 3
Recursive definition: Visit root after children
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v)
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18. Traversal example 7 Pre-order: 7 4 3 5 12 9 Post-order: 3 5 4 9 12 7 4
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19. Implementing trees List Tree class ListNode Node: data & reference
Node: data & references
class ListNode
{int data;
ListNode next; }
class TreeNode
TreeNode
child1,
child2,
child3;
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20. Implementing trees Problem of representing a node:
How many edges to allow for? e.g.,
Class TreeNode {
int data;
TreeNode
child1,
child2,
child3; }
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21. Implementing trees Problem of representing a node: Three solutions:
allow lots of child edge references
restrict tree structure to two child edges (binary tree)
use first child/next sibling representation
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22. First child/next sibling
Class TreeNode {
int data;
TreeNode
child, sibling; }
Problem:
Makes path to most nodes longer
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