1. Data Structures
Binary Trees 1

2. (Maximum level of the tree) leaves or nodes Level=1
Root Level = 0
number of leaves: ??
leaves
Depth
(Maximum level of the tree)
leaves or nodes
Level=1
height
Level=2
root 2
57 feet
Number of nodes: 2(2+1) - 1 = 7
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3. CIS265/506: Chapter 08 Intro to Binary Trees
Trees – Levels - Nodes
Max. number of leaves/nodes: 2(d+1) - 1 where d = depth (assume the tree is full (complete))
The level of a node is its distance from the root. The depth of a tree is the level of the leaf in the longest path from the root plus one.
A strictly binary tree: every non-leaf node has nonempty left & right sub trees
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4. CIS265/506: Chapter 08 Intro to Binary Trees
Trees – Levels - Nodes
A complete binary tree of depth d has all leaves at level d
An almost complete binary tree of depth d is such that:
Any node at a level less than (d-1) has 2 sons
For any node with a right descendant at level d, it must have a left son and every left descendant is either a leaf at level d or has 2 sons.
A tree can be strictly binary, but not almost complete
Conversely, a tree can be almost complete, but not strictly binary.
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5. CIS265/506: Chapter 08 Intro to Binary Trees
A binary tree is a tree whose nodes have children (possibly empty) and each child is either a left child or a right child.
Each circle is a node. The circle right below it (and attached by the arrow) is the child (or leaf). Each child is either the left or right child.
Each child can have only one parent, and each parent can have only two children.
Each path (from node to node) is a straight line. There are no cycles in binary trees.
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6. CIS265/506: Chapter 08 Intro to Binary Trees
This is a tree, but not a binary tree. There are 2 nodes that have three children.
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7. Are these binary trees? Why or why not??
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8. CIS265/506: Chapter 08 Intro to Binary Trees
Why do we have binary trees?
Improve speed when searching a long list
Trees are “naturally ordered” - information is placed in a logical location so it can be found easier. 1 2 3 4 5 3 4 2 5 1
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9. Implementing a Binary Tree
Trees can be implemented as either arrays or dynamic structures, similar to linked lists. We will only look at the dynamic implementation - the array implementation can be difficult to manage.
left
info
right
Left & right are pointers to nodes (children) in the tree. Info is where your data are stored.
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10. Conceptual View of a Tree
data
*left
*right
data
data
*left
*right
*left
*right
data
data
data
data
*left
*right
*left
*right
*left
*right
*left
*right
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11. Binary Trees - Ordering
Trees can be ordered in many ways. The most common are:
Hierarchical: Highest to lowest. Example - A university has multiple campuses, which have multiple colleges, which have many departments, etc.
Chronological: The order in the tree is determined by when the item was placed in the tree.
Search Trees (Alternately called ordered trees): A formal definition: if a node n of the tree contains the value v, then all the values stored in the left subtree are less than v, and all the values in the right subtree are greater than v. Duplicate values are not allowed.
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12. CIS265/506: Chapter 08 Intro to Binary Trees
Traversing a tree
Three important ways to visit every node of a tree.
INORDER Inorder visits the left child, root, then right child
PREORDER Preorder visits the root, the left child, and then
the right child
POSTORDER Postorder visits the left child, the right child,
and finally the root.
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13. CIS265/506: Chapter 08 Intro to Binary Trees
Tree Traversal
The mechanics of visiting nodes could be accomplished by either: an explicitly navigation (I will do everything by myself – approach), or using recursion (let the system help me).
The explicit method requires you to define and use a stack - all data points are pushed on the stack as we visit them, and we then pop each element from the stack to view the data. This could be tedious and expensive in terms of time & space. It also requires the additional data structure of at least one generic stack to do any tree manipulations.
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14. CIS265/506: Chapter 08 Intro to Binary Trees
Sample Tree Structure
Tree (A pointer to the root)
Left Sub Tree
Right Sub Tree
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15. CIS265/506: Chapter 08 Intro to Binary Trees
Examining One Node
tree
info
tree  left
tree  right
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16. CIS265/506: Chapter 08 Intro to Binary Trees
Tree Traversal
The other (more elegant) visitation method is based on recursion. This requires a bit of extra effort on your part when debugging, but the code is much shorter and simpler to write.
The algorithm for INORDER Traversals:
Assume the tree exists Visit the left subtree Visit the root Visit the right subtree
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17. CIS265/506: Chapter 08 Intro to Binary Trees
Let’s Look at the Code
// Inorder visits:
// the left child, root, then right child
void inOrderTraversal(NodePtr tree) { if (tree != NULL) { inOrderTraversal(tree->left); System.out.println(tree->info); inOrderTraversal(tree->right); } }
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18. CIS265/506: Chapter 08 Intro to Binary Trees
A Very Simple Case B
blue A C
red
black
Is the tree NULL? No, it does exist
“inOrderTraversal(tree->left)” - Visit the red (A) node: We call the procedure again using the red (A) node as the tree. There is no left node of red (A) , so we print the root. Now, there is no right node of red (A) , so we exit the recursive call.
“System.out.println(tree->info);” - We print the blue (B) node.
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19. A Simple Case (Continued)B
blue A C
red
black
“inOrderTraversal(tree->right)” - Visit the black (C) node: We call the procedure again using the black (C) node as the tree. There is no left node of black (C) , so we print the root. Now, there is no right node of black (C) , so we exit the recursive call.
“/* end if */” - We are done.
Our output is: red (A) blue (B) black (C)
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20. Let’s do one that is more difficultB E A D F
A B C D E F
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21. CIS265/506: Chapter 08 Intro to Binary Trees
PreOrder Traversal
Preorder visits the root, the left child, and then the right child
The algorithm for PREORDER Traversals:
Assume the tree exists Visit the root Visit the left subtree Visit the right subtree
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22. CIS265/506: Chapter 08 Intro to Binary Trees
Let’s Look at the Code
// Preorder visits
// the root, left child, then right child
void preOrderTraversal(NODEPTR tree) { if (tree != NULL) { System.out.println(tree->info); preOrderTraversal(tree->left); preOrderTraversal(tree->right); } }
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23. CIS265/506: Chapter 08 Intro to Binary Trees
A Very Simple Case B
blue A C
red
black
Is the tree NULL? No, it does exist
“System.out.println(tree->info);” - We print the blue (B) node.
“preOrderTraversal(tree->left)” - Visit the red (A) node: We call the procedure again using the red (A) node as the tree. We print the root (red (A) ) and note that there is no left node of red (A) or no right node of red (A), so we exit the recursive call.
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24. A Very Simple Case (Continued)B
blue A C
red
black
“preOrderTraversal(tree->right)” - Visit the black (C) node: We call the procedure again using the black (C) node as the tree. We print the root (black (C) ) and note that there is no left node of black (C) or no right node of black (C), so we exit the recursive call.
“/* end if */” - We are done.
Our output is: blue (B) red (A) black (C)
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25. CIS265/506: Chapter 08 Intro to Binary Trees
Let’s do one that is more difficult C B E A D F
C B A E D F
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26. CIS265/506: Chapter 08 Intro to Binary Trees
PostOrder Traversal
Postorder visits the the left child, the right child, then the root
The algorithm for POSTORDER Traversals:
Assume the tree exists Visit the left subtree Visit the right subtree Visit the root
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27. CIS265/506: Chapter 08 Intro to Binary Trees
Let’s Look at the Code
// Postorder visits
// left child, right child, then the root
void postOrderTraversal(NODEPTR tree) { if (tree != NULL) { postOrderTraversal(tree->left); postOrderTraversal(tree->right); System.out.println(tree->info); } }
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28. CIS265/506: Chapter 08 Intro to Binary Trees
A Very Simple Case B
blue A C
red
black
Is the tree NULL? No, it does exist
“postOrderTraversal(tree->left)” - Visit the red (A) node: We call the procedure again using the red (A) node as the tree. We note that there is no left node of red (A) or no right node of red, so we print the root (red (A) ) and exit the recursive call.
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29. CIS265/506: Chapter 08 Intro to Binary Trees
A Very Simple Case (Continued) B
blue A C
red
black
“postOrderTraversal(tree->right)” - Visit the black (C) node: We call the procedure again using the black (C) node as the tree. We note that there is no left node of black (C) or no right node of black (C), so we print the root (black (C) ) and exit the recursive call.
“System.out.println(tree->info);” - We print the blue(B) node.
“/* end if */” - We are done.
Our output is: red (A) black (C) blue (B)
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30. CIS265/506: Chapter 08 Intro to Binary Trees
Let’s do one that is more difficult C B E A D F
A B D F E C
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