department of computer engineering faculty of engineering, prince of songkla university 1 8 –...
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Department of Computer Engineering Faculty of Engineering, Prince of Songkla University
8 – Trees8 – Trees
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Overview:
1. Tree Example
2. Binary Trees
3. Binary Trees in C
4. Tree Traversal
5. Binary Search Trees
6. Searching a BST
7. Insertion into a BST
8. Deletion from a BST
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1. Tree Example
Board of Directors
ResearchDivision
ManufacturingDivision Admin Marketing
ComputerSystems
AccountsPayable Payroll
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2. Binary Trees
Each element (node) of the tree may have 0, 1 or 2 successors
a
c
gfed
b
kjih
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Binary Tree Jargon
a is the root of the tree
b, c are the children of a
left subtree of a: the tree whose root is b
right subtree of a: the tree whose root is c
d, e, h, i, j are descendents of b
every node is a descendent of a (the root)
h, i, j, k, g are the leaves of the tree
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Shape
Examples for 6 nodes:
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Some Binary Tree ADT Operations
create an empty tree
build a new tree from a data value and two subtrees
deletion
insertion
check to see if a tree is empty
print out all the elements in a tree
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3. Binary Trees in C
3.1 Binary Tree Type
3.2 Examples
3.3 Some Functions
3.4 Size of a Binary Tree
3.5 Height of a Binary Tree
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3.1 Binary Tree Type
struct treenode { struct treenode *leftptr; int data;
struct treenode *rightptr;
};
typedef struct treenode TREENODE; typedef TREENODE *TREE;
treenodedata
rightptrleftptr
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3.2 Examples
TREE t;
The empty tree:
t = NULL;
A tree with a single element 2:
t
NULLt
t 2
N N
1111
t 2
4
7
1
N N N
NN
A more complicated tree:
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3.3 Some Functions
int isEmptyTree ( TREE t ){ if (t == NULL) return 1; /* true */ else return 0; /* false */ }
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int value(TREE t){ if (isEmptyTree(t)) { printf(“Tree is empty \n”); return -1; } else return t->data;}
TREE mkEmpty(void)/* return an empty tree */{ return NULL;}
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TREE mkTree(int x, TREE leftT,TREE rightT)/* make a new tree from a data value x and two existing trees */{ TREE temp;
temp = malloc(sizeof(TREENODE)); temp->data = x; temp->leftptr = leftT; temp->rightptr = rightT;
return temp;}
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TREE lsTree(TREE t){ return t->leftptr; /* don’t bother dealing with the empty tree case */}
TREE rsTree(TREE t){ return t->rightptr; }
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3.4 Size of a Binary Tree
an empty tree has 0 size
a non-empty tree has one node plus the number of elements in the subtrees
n
tree2tree1
size(tree) = 1 + size(tree1) +
size(tree2)
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C Version
int size(TREE t){ if (isEmptyTree(t)) return 0; else return (1 + size(lsTree(t))
+ size(rsTree(t)) );}
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Rephrased
int size(TREE t){ if (t == NULL) return 0; else return (1 + size(t->leftptr)
+ size(t->rightptr)); }
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3.5 Height of a Binary Tree
the height of an empty tree is 0
the height of a non-empty tree is 1 plus the height of the highest subtree
n
tree2tree1
heightof tree1 height
of tree2
height(tree) = 1 + max(height(tree1), height(tree2))
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C Version
int height(TREE t){ if(isEmptyTree(t)) return 0; else return (1 + max(height(lsTree(t)),
height(rsTree(t))));}
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2 general approaches to traversal sequence
depth-first traversal proceeds along a path from the root through
one child to the most distant descendent of that first child before processing a second child
breadth-first traversal proceeds horizontally from the root to all of its
children, then to its children’s children, and so forth until all nodes have been processed
4. Tree Traversal
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Depth-First Traversals
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3 different depth-first traversal sequences
1. Preorder Traversal (N-L-R) look at the root node first, then the left subtre
e, then the right subtree
2. Inorder Traversal (L-N-R) (or mid-order) look at the left subtree first, then the root
node, then the right subtree
3. Postorder Traversal (L-R-N) look at the left subtree first, then the right su
btree, then the root node
4.1 Depth-First Traversals
N
RL
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Traversal Nodes Visited preorder. . . . . . 1 2 4 8 5 9 10 3 6 11 7 inorder. . . . . . . 8 4 2 9 5 10 1 6 11 3 7 postorder. . . . . 8 4 9 10 5 2 11 6 7 3 1
1
3
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2
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Example
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Print a Tree using Preorder
void preorder_print(TREE t){ if (!isEmptyTree(t)) { printf(“%3d”, value(t));
preorder_print(lsTree(t)); preorder_print(rsTree(t)); }}
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Inorder and Postorder Printing
void inorder_print(TREE t){ if (!isEmptyTree(t)) { inorder_print(lsTree(t)); printf(“%3d”, value(t));
inorder_print(rsTree(t)); }}
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void postorder_print(TREE t){ if (!isEmptyTree(t)) { postorder_print(lsTree(t)); postorder_print(rsTree(t)); printf(“%3d”, value(t)); }}
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all of the children of a node before proceeding with the next levelthat is, given a root at level n, we process all nodes at level n before processing with the nodes at level n + 1to traverse a tree in depth-first order, we use a stack (Remember: recursion uses stack )to traverse a tree in breadth-first order, we use a queue
4.2 Breadth-first Traversal
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5. Binary Search Trees
5.1 Informal Definition
5.2 Examples
5.3 C BST Type
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5.1 Informal Definition
A Binary Search Tree (BST) is a binary tree, with an ordering on its nodes.
In every subtree (including the whole tree): all values in the left subtree are 'less than' the
value in the root node all values in the right subtree are 'greater than'
the value in the root node
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5.2 Examples
4
6
5 731
2
4
6
5 713
2
is a BST
is not a BST
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5.3 C BST Type
Change the typedef names to reflect the fact that we are treating the tree as a BST:
typedef struct treenode BSTNODE;typedef BSTNODE *BST;
C’s typing is not powerful en ough to
include the BST ordering information
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6. Searching a BST
6.1 Informal Definition
6.2 Example
6.3 C Version
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6.1 Informal Definition
To recursively search for an item (called the key) in a BST:
A key cannot be in an empty tree
For a non-empty tree: if key == root value, stop if key < root value, search left subtree if key > root value, search right subtree
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6.2 Example
Consider searching for 5
4
6
5 731
2
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6.3 C Version
int bstsearch(int key, BST t){ int v; if (isEmptyTree(t)) return 0; v = value(t); if (key == v) return 1; if (key < v)
return bstsearch(key, lsTree(t)); if (key > v) return bstsearch(key, rsTree(t));}
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7. Insertion into a BST
7.1 Examples
7.2 Informal Insertion Definition
7.3 Tree Creation Functions
7.4 bstinsert()
7.5 More Efficient Insertion
7.6 Comparing bstinsert() and insert_node()
7.7 insert_node2()
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7.1 Examples
Insert the following into an empty BST:
4 2 3 6 1 5 7
The resulting tree:
4
6
5 731
2
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Rearrange the values, and reinsert them:
5 7 1 4 6 2 3
The resulting tree:
4 6
5
7
3
1
2
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7.2 Informal Insertion Definition
1. If the tree is empty, the value becomes the root of a new tree.
If the tree is not empty, then:
2. If value < root, it must be inserted into the left subtree.
3. If value > root, it must be inserted into the right subtree.
4. If value == root, then stop.
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7.3 Tree Creation Functions
BST mkEmpty(void){ return NULL;}
BST mkTree(int x,BST leftT, BST rightT){ BST t; t = malloc(sizeof(BSTNODE)); t->data = x; t->leftptr = leftT; t->rightptr = rightT; return t;}
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7.4 bstinsert()
BST bstinsert(int x, BST t){
int v; BST t1, t2;
if (isEmptyTree(t)) return mkTree(x,mkEmpty(),mkEmpty()); v = value(t); //value of root node t1 = lsTree(t); t2 = rsTree(t);
continued
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if (x < v) return mkTree(v,bstinsert(x,t1),t2); else if (x > v) return mkTree(v,t1,bstinsert(x,t2)); else return mkTree(v, t1, t2); /* or return t; */}
• easy to understand
• inefficient due to copying
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Use:
BST t, newt;
t = bstinsert(5, bstinsert(7, bstinsert(1, bstinsert(2, bstinsert(4, mkEmpty())))));
newt = bstinsert(9, t);
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4
5
7
91
2
t
4
7
newt
bstinsert() has copied all of the n odes in the path from the root dow
n to the parent node of where the number has been inserted.
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7.5 More Efficient Insertion
Modify the input tree to include the new value.
no copying
destructive assignment
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insert_node()
void insert_node(BST *tp, int value){ /* tp is a pointer to a BST */ if( *tp == NULL) {
*tp = malloc(sizeof(BSTNODE));
(*tp)->data = value;
(*tp)->leftptr = NULL;
(*tp)->rightptr = NULL; } else if (value < (*tp)->data) insert_node(&((*tp)->leftptr),value); else if (value > (*tp)->data) insert_node(&((*tp)->rightptr),value); else printf(“duplicate node\n”);
}
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Use:
BST t = NULL;
insert_node(&t, 4); /* address of pointer */
insert_node(&t, 2);
insert_node(&t, 5);
insert_node(&t, 7);
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What is &((*tp)->leftptr)?
tp is a pointer to a BST
*tp is a BST
(*tp)->leftptr is the left subtree, another BST
&((*tp)->leftptr) is the address of that BST, which is the type of the first arg of insert_node()
The same reasoning applies to the use of: &((*tp)->rightptr)
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7.6 Comparing bstinsert() and insert_node()
bstinsert() easy to understand but inefficient copying
insert_node() efficient but hard to understand call by reference
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7.7 insert_node2()
/* call by reference replaced by returns */
BST insert_node2(BST t, int value){ if (t == NULL) { t = malloc(sizeof(BSTNODE)); t->data = value; t->leftptr = NULL; t->rightptr = NULL; }
continued
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else if (value < t->data) t->leftptr = insert_node2(t->leftptr, value);
else if (value > t->data) t->rightptr = insert_node2(t->rightptr, value);
else printf("duplicate node\n");
return t;}
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Use:
BST t = NULL;
t = insert_node2(t, 4);
t = insert_node2(t, 2);
t = insert_node2(t, 5);
t = insert_node2(t, 7);
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8. Deletion from a BST
8.1 Deletion Cases
8.2 Examples
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8.1 Deletion Cases
There are three cases. The deleted node has:
1) no children
2) one child
3) two children
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8.2 Examples
Consider the BST t:
4 6
5
8
3
1
2 7
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Case 1: deleted node has No Children
Delete 3 from t:
4 6
5
8
3
1
2 7
4 6
5
81
2 7
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Case 2 : deleted node has 1 Child
Delete 8 from t:
4
6
5
1
74 6
5
81
2 7
3 • adjust the link to the child of the deleted node
2
3
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Case 3: deleted node has 2 Children
n
tree2tree1
rest oftree
delete n
x
x
tree2tree1
rest oftree
3.1) Select the least value in the right subtree to replace x is the next biggest value after n
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Example: delete 3 (and replaced by node from the right)
8
3
5
7
4
1
85
7
4
1
2 2
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Case 3: deleted node has 2 Children
n
tree1 tree2
rest oftree
delete n
y
y
tree2tree1
rest oftree
3.2) Select the greatest value in the left subtree to replace y is the biggest value less than n
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Example: 3delete (and replaced by node from the left)
8
3
5
7
4
1
2 8
2
5
7
4
1