Held Monday, April 24, 2000
Today, we generalize the heap and the search tree into a generic
tree data structure.
- I was hoping to grade assignment 5 over the weekend. Unfortunately, as
in all holiday weekends, the time evaporated.
- Are there project questions?
- Heaps and search trees, revisited
- Representing arithmetic expressions
- Generalizing: a binary tree ADT
- Generalizing further: a tree ADT
- Until recently, all of the ``ordered'' structures we considered
(lists, arrays, vectors, stacks, and queues) were somewhat linear
(in an informal rather than a formal sesnse).
- What makes a structure linear?
- Exactly one first element.
- Exactly one last element.
- Each element (except the first) has exactly
- Each element (except the last) has exactly
- Search trees and heaps had a somewhat different structure. How are
search trees Different?
- There is still exactly one first element.
- There are now many last elements.
- Each element (except the first) still has exactly
- Each element (except the last) may have
many (okay, two) successors.
- By careful consideration of such restrictions, we can develop a number
of different interesting structures.
- We can generalize search trees to develop a data structure commonly called
- Trees are a collection of values,
- stored in nodes,
- with a designated root (first element)
- and a number of leaves (``last'' elements).
- Every non-leaf node has one or more children
- and the leaves have zero children,
- Every nonroot node has exactly one parent (previous
- and the root has no parents.
- In binary trees, each node can have at most two children,
often designated as left and right.
- There are many ways in which trees are used, including,
- search trees, which are used to store information for retrieval;
- decision trees, which are used to describe decision processes;
- class hierarchies, which show the relationships between classes
in a single-inheritance language like Java;
- representing arithmetic expressions, using operators for internal
nodes and values for leaves; and even
- program trees, which show the structure of a program.
- You've already seen binary search trees:
- A node stores a value.
- All smaller values are in the left subtree.
- All larger values are in the right subtree.
- You've also seen heaps:
- The root stores the smallest value.
- The left and right subtrees are also heaps.
- The tree is nearly-complete.
- Trees are good for hierarchical structures, like advising relationships
in graduate school.
- Purposefully left open. I'm going to ask you to design this one.
- The basics so far:
public interface Tree
* Add another child to a vertex in the tree.
* Pre: The vertex (parent) is in the current tree.
* Post: The vertex (parent) has another child. The new child
* contains newThing.
public void add(Object newThing, Vertex parent)
* Get the root of the tree.
public _______ getRoot();
Tuesday, 18 January 2000
- Created as a blank outline.
Monday, 24 April 2000
Tuesday, 25 April 2000
- Updated to better represent what happened.
Back to Project Discussion.
On to Implementing Trees.