1. Trees
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2. Tree Basics A tree is a set of nodes.
A tree may be empty (i.e., contain no nodes).
If not empty, there is a distinguished node, r, called the root and zero or more non-empty subtrees T1, T2, T3,….Tk, each of whose roots is connected by a directed edge from r.
Trees are recursive in their definition and, therefore, in their implementation.
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3. A General Tree A B G K L C D H I J E F
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4. Tree Terminology
Tree terminology takes its terms from both nature and genealogy.
A node directory below the root, r, of a subtree is a child of r, and r is called its parent.
All children with the same parent are called siblings.
A node with one or more children is called an internal node.
A node with no children is called a leaf or external node.
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5. Tree Terminology (con’t)
A path in a tree is a sequence of nodes, (N1, N2, … Nk) such that Ni is the parent of Ni+1 for 1 <= i <= k.
The length of this path is the number of edges encountered (k – 1).
If there is a path from node N1 to N2, then N1 is an ancestor of N2 and N2 is a descendant of N1.
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6. Tree Terminology (con’t)
The depth of a node is the length of the path from the root to the node.
The height of a node is the length of the longest path from the node to a leaf.
The depth of a tree is the depth of its deepest leaf.
The height of a tree is the height of the root.
True or False – The height of a tree and the depth of a tree always have the same value.
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7. Tree Storage First attempt - each tree node contains
The data being stored
We assume that the objects contained in the nodes support all necessary operations required by the tree.
Links to all of its children
Problem: A tree node can have an indeterminate number of children. So how many links do we define in the node?
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8. First Child, Next Sibling
Since we can’t know how many children a node can have, we can’t create a static data structure -- we need a dynamic one.
Each node will contain
The data which supports all necessary operations
A link to its first child
A link to a sibling
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9. First Child, Next Sibling Representation
To be supplied in class
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10. Tree Traversal
Traversering a tree means starting at the root and visiting each node in the tree in some orderly fashion.
The purpose of traversing a tree is to enable us to perform operations such as:
print all data stored in the tree
search for a particular data item in the tree
compute the sum of all corresponding data items in the tree (e.g., sum all student exam scores where there is one score per node)
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11. Breadth-First Tree Traversals
Start at the root.
Visit all the root’s children.
Then visit all the root’s grand-children.
Then visit all the roots great-grand-children, and so on.
This traversal goes down by levels.
A queue can be used to implement this algorithm.
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12. BF Traversal Pseudocode
Create a queue, Q, to hold tree nodes
Q.enqueue (the root)
while (the queue is not empty) Node N = Q.dequeue( ) for each child, X, of N Q.enqueue (X)
The order in which the nodes are dequeued is the BF traversal order.
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13. Depth-First Traversal
Start at the root.
Choose a child to visit.
Visit all of that child’s children.
Visit all of that child’s children’s children, and so on.
This traversal goes down a path until the end, then comes back and does the next path.
A stack can be used to implement this algorithm.
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14. DF Traversal Pseudocode
Create a stack, S, to hold tree nodes
S.push (the root)
While (the stack is not empty) Node N = S.pop ( ) for each child, X, of N S.push (X)
The order in which the nodes are popped is the DF traversal order.
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15. Performance of BF and DF Traversals
What is the asymptotic performance of breadth-first and depth-first traversals on a general tree?
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16. Binary Trees
A binary tree is a tree in which each node may have at most two children and the children are designated as left and right.
A full binary tree is one in which each node has either two children or is a leaf.
A perfect binary tree is a full binary tree in which all leaves are at the same level.
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17. A Binary Tree
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18. A binary tree?
A full binary tree?
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19. A binary tree? A full binary tree? A perfect binary tree?
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20. Binary Tree Traversals
Because nodes in binary trees have at most two children (left and right), we can write specialized versions of BF and DF traversals. These are called
In-order traversal
Pre-order traversal
Post-order traversal
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21. In-Order Traversal At each node visit my left child first visit me
visit my right child last 8 5 3 9 12 7 15 6 2 10
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22. In-Order Traversal Code
void inOrderTraversal(Node *nodePtr) {
if (nodePtr != NULL) {
inOrderTraversal(nodePtr->leftPtr);
cout << nodePtr->data << endl;
inOrderTraversal(nodePtr->rightPtr); }
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23. Pre-Order Traversal At each node visit me first
visit my left child next
visit my right child last 8 5 3 9 12 7 15 6 2 10
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24. Pre-Order Traversal Code
void preOrderTraversal(Node *nodePtr) {
if (nodePtr != NULL) {
cout << nodePtr->data << endl;
preOrderTraversal(nodePtr->leftPtr);
preOrderTraversal(nodePtr->rightPtr); }
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25. Post-Order Traversal At each node visit my left child first
visit my right child next
visit me last 8 5 3 9 12 7 15 6 2 10
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26. Post-Order Traversal Code
void postOrderTraversal(Node *nodePtr) {
if (nodePtr != NULL) {
postOrderTraversal(nodePtr->leftPtr);
postOrderTraversal(nodePtr->rightPtr);
cout << nodePtr->data << endl; }
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27. Binary Tree Operations
Recall that the data stored in the nodes supports all necessary operators. We’ll refer to it as a “value” for our examples.
Typical operations:
Create an empty tree
Insert a new value
Search for a value
Remove a value
Destroy the tree
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28. Creating an Empty Tree Set the pointer to the root node equal to NULL.
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29. Inserting a New Value The first value goes in the root node.
What about the second value?
What about subsequent values?
Since the tree has no properties which dictate where the values should be stored, we are at liberty to choose our own algorithm for storing the data.
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30. Searching for a Value
Since there is no rhyme or reason to where the values are stored, we must search the entire tree using a BF or DF traversal.
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31. Removing a Value
Once again, since the values are not stored in any special way, we have lots of choices. Example:
First, find the value via BF or DF traversal.
Second, replace it with one of its descendants (if there are any).
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32. Destroying the Tree
We have to be careful of the order in which nodes are destroyed (deallocated).
We have to destroy the children first, and the parent last (because the parent points to the children).
Which traversal (in-order, pre-order, or post-order) would be best for this algorithm?
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33. Giving Order to a Binary Tree
Binary trees can be made more useful if we dictate the manner in which values are stored.
When selecting where to insert new values, we could follow this rule:
“left is less”
“right is more”
Note that this assumes no duplicate nodes (i.e., data).
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34. Binary Search Trees
A binary tree with the additional property that at each node,
the value in the node’s left child is smaller than the value in the node itself, and
the value in the node’s right child is larger than the value in the node itself.
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35. A Binary Search Tree 50 57 42 67 30 53 22 34
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36. Searching a BST Searching for the value X, given a pointer to the root
If the value in the root matches, we’re done.
If X is smaller than the value in the root, look in the root’s left subtree.
If X is larger than the value in the root, look in the root’s right subtree.
A recursive routine – what’s the base case?
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37. Inserting a Value in a BST
To insert value X in a BST
Proceed as if searching for X.
When the search fails, create a new node to contain X and attach it to the tree at that node.
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38. Inserting
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39. Removing a Value From a BST
Non-trivial
Three separate cases:
node is a leaf (has not children)
node has a single child
node has two children
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40. Removing
100
150 50 70
120 30
130 20 40 60 80 85 55 65 53 57
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41. Destroying a BST
The fact that the values in the nodes are in a special order doesn’t help.
We still have to destroy each child before destroying the parent.
Which traversal can we use?
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42. Performance in a BST What is the asymptotic performance of:
insert
search
remove
Is the performance of insert, search, and remove for a BST improved over that for a plain binary tree?
If so, why?
If not, why not?
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