1. Cs212: Data Structures Lecture 7: Tree_Part1
Computer Science Department
Lecture 7: Tree_Part1

2. Trees Linked lists usually are more flexible than arrays,
but it is difficult to use them to organize a hierarchical representation of objects.
Stacks and queues reflect some hierarchy but are limited to only one dimension.
For these reasons, create a new data type tree that consists of nodes and arcs/edges and has the root at the top and the leaves (terminal nodes) at the bottom.
21-Dec-19
Computer Science Department

3. Tree Definitions and Properties
A tree is an abstract data type that stores elements hierarchically.
With the exception of the top element, each element in a tree has a parent element and zero or more children elements.
Formally, we define a tree T as a set of nodes storing elements such that the nodes have a parent-child relationship, that satisfies the following properties:
If T is nonempty, it has a special node, called the root of T, that has no parent.
Each node v of T different from the root has a unique parent node w; every node with parent w is a child of w.

4. Tree Terminology / concepts
Empty tree: meaning that it doesn't have any nodes.
Parent : Any node (except the root) has exactly one edge running upward to another node or
A node is a parent if it has successor nodes; that is, if it has outdegree greater than zero.
Child : Any node can have one or more lines running downward to other nodes.
It has a predecessor node
Siblings :Two or more nodes with the same parent.
A node v is external if v has no children. External nodes are also known as leaves.
A node v is internal if it has one or more children.
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Computer Science Department

5. Example
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6. Tree Terminology / concepts con..
Ancestor : Any node in the path from the root to the node
Descendent :
Any node in the path below the parent node
All nodes in the paths from a given node to a leaf node
Subtree : any node can be considered to be the root of a subtree, which consists of its children, and its children's children, and so on.
An edge of tree T is a pair of nodes (u, v) such that u is the parent of v, or vice versa.
A path of T is a sequence of nodes such that any two consecutive nodes in the sequence form an edge.
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Computer Science Department

7. Tree Terminology / concepts con..
Level of the node :
The length of the path from the root to the node.
Level of a node is its distance from the root.
Length : the number of arcs in a path.
Visiting : a node is visited when program control arrives at the node, usually for the purpose of carrying out some operation on the node.
Traversing : to visit all the nodes in some specified order.
Key : a value that is used to search for the item or perform other operations on it.

8. Tree representing a portion of a file system.
cs252/ is an ancestor of papers/, and pr3 is a descendent of cs016/.
The subtree rooted at cs016/ consists of the nodes cs016/, grades, homeworks/, programs/, hw1, hw2, hw3, pr1, pr2, and pr3.
the tree contains the path (cs252/, projects/, demos/, market).
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Computer Science Department

9. Ordered Trees
A tree is ordered if there is a linear ordering defined for the children of each node. that is, we can identify the children of a node as being the first, second, third, and so on.
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Computer Science Department

10. The Tree Abstract Data Type
The tree ADT stores elements at positions, which, as with positions in a list, are defined relative to neighboring positions.
The positions in a tree are its nodes, and neighboring positions satisfy the parent-child relationships that define a valid tree.
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Computer Science Department

11. The Tree Abstract Data Type con..
a position object for a tree supports the method : element(): return the object stored at this position. root(): return the tree's root; an error occurs if the tree is empty. parent (v): return the parent of v; an error occurs if v is the root.
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Computer Science Department

12. The Tree Abstract Data Type con..
children(v): return an iterable collection containing the children of node v. If a tree T is ordered, then the iterable collection, children(v), stores the children of v in order. If v is an external node, then children(v) is empty.
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Computer Science Department

13. The Tree Abstract Data Type con..
Query methods: isInternal(v): Test whether node v is internal. isExternal(v): Test whether node v is external. isRoot(v): Test whether node v is the root.
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Computer Science Department

14. The Tree Abstract Data Type con..
Generic methods: size(): return the number of nodes in the tree. isEmpty(): Test whether the tree has any nodes or not. iterator(): return an iterator of all the elements stored at nodes of the tree. positions(): return an iterable collection of all the nodes of the tree. replace(v,e): Replace with e and return the element stored at node v.
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15. Implementing a Tree
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