comp 245 data structures - harding 7 - trees.pdf · traversing a tree pre –order (v ... binary...
TRANSCRIPT
Introduction to the Tree ADT
A tree is a non-linear structure.
A treenode can point to 0 to N other
nodes.
There is one access point to the tree;
it is called the root.
A tree is recursive in nature.
Traversing a Tree
PRE – order
(v)isit – (l)eft – (r)ight VLR
POST – order
(l)eft – (r)ight – (v)isit LRV
IN – order
(l)eft – (v)isit – (r)ight LVR
Sample Code of a Traversal
void Tree::InOrder(TreePtr P)
{
if (P != NULL)
{
InOrder(P->Left);
Process(P);
InOrder(P->Right);
}
}
Implementing a Binary Tree
Linked
struct Node;
typedef SomeDataType TreeType;
typedef Node* TreePtr;
struct Node{
TreeType info;TreePtr Left, Right;
};
Defining a Binary Tree
Linked
class Tree
{
public:
Tree();
~Tree();
bool Empty();
bool Insert(TreeType);
bool Delete(TreeType);
void Traverse();
private:
void InOrder(TreePtr);
TreePtr Root;
};
Binary Search Trees
BST
A special type of tree that is very
useful!
Main characteristic:Given any node P, the left child is lesser than or
equal to P; the right child is greater than P.
The efficiency of a BST ranges from
logarithmic time to linear time.
BST Efficiency
Assuming a tree is balanced, it’s efficiency is approximately log2N where N is the number of elements in the tree.
Example:There are 1000 elements in a BST, it’s efficiency therefore
is approximately log21000 = 9.9 or 10. This means that it will take in the absolute worst case, 10 accesses to find a value in the tree. If you contrast this to an ordered list, it will take 1000 accesses in the worst case and 500 in the average case to find an element!!
If a tree is not balanced, it’s efficiency will degenerate!
BST Operation - Insertion
The Insert function can be highly
efficient.
The new value is always inserted as a
leaf node!
BST Operation – Insertion
Practice: Build a BST
Build a BST from these values:
LARRY
FRED
JOE
STEVE
NANCY
BILL
CAROL
TERRY
Inserting a Node into a BST
Create a node (Test for success)
Store data, set right and left pointers
null (it will be a leaf)
Search tree for insertion point, keep
track of node which will become the
parent.
Attach this node to parent.
Return success or failure of operation.
Deleting a Node from a BST
There are three cases to account for:Leaf
One Child
Two Child
The algorithm requires a Search to find the node to delete, determining the specific case, and then executing the deletion.
Leaf Case
How do you know the node is a leaf?
This routine will require 1) a pointer to the node to
be deleted and 2) a pointer to the parent.
Delete Leaf Code
void Bst::DeleteLeaf (TreePtr P, TreePtr Parent)
{
//check for root
if (Parent == NULL)
Root = NULL;
else
if (Parent->Left == P)
Parent->Left = NULL;
else
Parent->Right = NULL;
delete P;
}
One Child Case
How do you know the node has one child?
This routine will require 1) a pointer to the node to be
deleted and 2) a pointer to the parent.
Delete One Child Code
void Bst::DeleteOneChild (TreePtr P, TreePtr Parent)
{
1) save pointer to subtree, must be re-attached
2) check for root case
3) re-attach subtree to parent
4) delete P
}
Two Child Case
How do you know the node has two children?
This routine will require only a pointer to the node to be
deleted.
Finding the Closest
Predecessor
From the two child node to be deleted, take one step left and go as far right as possible. This node is the closest predecessor.
Place this value in the node to be deleted.
The closest predecessor will be deleted by calling DeleteLeaf or DeleteOneChild.
Delete Two Children Case
void Bst::DeleteTwoChild (TreePtr P)
{
1) Find closest predecessor (cp), keep track of
parent to cp!!
2) Copy cp->info to P->info
3) Call DeleteLeaf or DeleteOneChild for cp
}
Traversal Usage
Preorder
The preorder traversal can be used to effectively
save a tree to file that can be reconstructed
identically. This type of traversal can be used to
copy a tree also.
MikeDonHarryGregTimPaulWayne
Traversal Usage
Inorder
The inorder traversal can be used to obtain a sorted
list from a BST.
DonGregHarryMikePaulTimWayne
Traversal Usage
Postorder
The postorder traversal can be used to delete a
tree. A tree needs to be deleted from the bottom up
because every node at the point of deletion is a
leaf.
Order of DeletionGregHarryDonPaulWayneTimMike
Binary Tree Implementation
Array Based – method 1
The first method will store information in the tree traveling down levels going left to right.
Given this storage technique, a node stored at slot I in the array will have it’s left child at 2I + 1, and the right child will be at 2I + 2.
A parent can be found at (I – 1)/2.
Binary Tree Implementation
Array Based – method 2
The second method will have an array
of structs. Each struct will contain the
information and left and right pointer
fields. The pointer fields will simply be
index values within the array.
Each new value is added at the end of
the array as a leaf and the pointer to
it’s parent adjusted.