1. CMSC 202
Trees

2. Tree Definitions A tree is a set of nodes
The set of nodes may be empty
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.
As item #3 indicates, trees are recursive in their definition and therefore in their implementation.

3. Basic Tree Terminology
Tree terminology takes its terms from both nature and genealogy.
The root of each subtree is a child of r, and r is called its parent.
A node with one or more children is called an internal node.
All children of the same internal node are called siblings.
A node with no children is called a leaf or external node.

4. Path in a Tree
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.

5. Height/Depth of a Node
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

6. Height/Depth of a Tree
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

7. Tree Storage First attempt - each tree node contains
The data element 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?

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 object which supports all necessary operations
A link to its first child
A link to a sibling

9. A B D C E H I J G F
A general tree

10. First Child, Next Sibling Representation
To be supplied in class

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, and so on
This traversal goes down by levels
A queue application

12. BF Traversal pseudo-code
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

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
Etc., etc., etc.
This traversal goes down a path until the end, then comes back and does the next path
A stack application

14. DF Traversal Pseudo-Code
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

15. Performance of BF and DF traversals
What is the asymptotic performance of Breadth-first and Depth-first traversals on a general tree?
Since the traversals “visit” every node in the tree, their performance is O(n), in all cases, where n is the number of nodes in the tree

16. Other functions
These fundamental functions could be written recursively with few lines of code
CountNodes – given a pointer to the root of a tree, count the number of nodes in the tree
Depth – given a pointer to the root of a tree, determine the tree’s depth

17. Binary Tree Definitions
A binary tree is a rooted tree in which each node has just 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

18. A Binary Tree

19. A Binary Tree?
A full Binary Tree?

20. A Binary Tree?
A full Binary Tree?
A perfect Binary Tree?

21. Binary Tree Traversals
Because nodes in binary trees have just 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
Level-order traversal

22. In-Order traversal At each node
“visit” my left child first
“visit” me
“visit” my right child last
A recursive function should you be able to write

23. Post-Order traversal At each node
“visit” my left child first
“visit” my right child next
“visit” me last
A recursive routine you should be able to write

24. Pre-Order Traversal At each node
“visit” me last first
“visit” my left child next
“visit” my right child last
A recursive routine you should be able to write

25. Binary Tree Operations
Recall that the object stored in the nodes support all necessary operators. We’ll refer to it as a “value” and use integers for our examples
Create an empty tree
Insert a new value
Search (find) a value
Remove a value
Destroy the tree

26. Creating an Empty Tree
Set the pointer to the root equal to NULL

27. Inserting into a Binary Tree
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

28. Searching in a Binary Tree
How do we find a value in a binary tree?
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

29. Removing a value from the tree
Once again, since the values are not stored in any special way, we have lots of choices
First, find the value via BF or DF traversal
Second, replace it with one of its children (if there are any).

30. Destroying a Tree
We have to be careful of the order in which nodes are destroyed
We have to destroy the children first, and the parent last (because the parent points to the children)

31. 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, follow this rule:
“left is less”
“right is more”

32. The Binary Search Tree Definition
A Binary Search tree is 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

33. 50 57 42 67 30 53 22 34
A Binary Search tree

34. 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?

35. Inserting a new 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

36. Removing a value from a BST
Non-trivial
We just can’t replace the node containing X with one of it’s children.
Why not?
What do we do instead?
There are three cases to examine

37. 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?

38. Performance in a BST
Is the performance of insert, search and remove improved?
If so, why?
If not, why not?