Recursive Definition of Tree Structures not so precise

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Recursive Definition of Tree Structures (not so
precise)
Ch 10 Trees
No rule is violated but we dont like it

• An empty collection is a tree
• A single node is a tree
• A node being linked to a finite number of trees
  forms a tree.

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Is this definition good enough ?
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Depict the Tree Structure in a better way
Terminologies come to play
The root
the links are directed
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height of d9 4
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height of d10 1
Ancestors
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leaf
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Parent
Descendants
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depth of d82
Children
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leaf
leaf
leaf
leaf
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leaf
Siblings
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More Terminologies

• In-degree the number of links pointing to a node
  (always 1 or 0) the root is the only node with
  in-degree 0.
• 2. Out-degree the number of link pointing out
  from a node leaves are node with out-degree 0
• 3 . Degree of a tree (arity) the maximum
  out-degree of nodes in the tree

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Recursive Definition of Tree Structures

• Empty is a tree the root is null
• A single node is a tree the node is the root of
  the tree.
• A node points to a finite number of the roots of
  some other trees form a tree the node is the
  root of the tree. (so, the node cant point to a
  node that is not the root.)

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Tree Implementation linked lists ?? arrays

• Linked Lists

data

• data field
• enough fields for pointers pointing the children

Usually, we use the degree of the tree as the
number of the pointer fields. Fixed, if the
degree is small, e.g., binary tree (degree is 2).
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Tree Implementation
Using Array

• Array is a good choice if
• the degree is small
• the tree is rather full
• the size does not change too often

height 3 degree 2
size 2height - 1
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What is the advantage and disadvantage of using
array?
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1st child (left-child) of i is 2i1 2nd child
(right-child) of i is 2i2. The parent of i is
(i-1)/2
height 4
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A full-tree or complete tree is a perfect
example for using array
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complete tree
full-tree
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A random binary tree
Randomly insert data into a binary tree
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A Binary Tree Node in Java

• A data field
• Two pointers to the left-child and right-child

public class BinaryTree ltE extends ComparableltEgtgt
/ This is an inner class for tree
nodes/ private static class TNodeltEgt
private E data private TNodeltEgt
left,right private TNode(E data, TNodeltEgt left,
TNodeltEgt right) //Construct a node with two
children this.data data this.left
left this.right right /
This is the end of the inner class NodeltEgt
/ private TNodeltEgt root public
BinaryTree() root null .....
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Add a new data to the BinaryTree
//add data to this BST, return false if data is
already in the tree public boolean add(E data)
if (root ! null) return add(root,
data) root new TNodeltEgt(data, null,
null) return true
A private overloaded add method root is private
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Randomly Insert (add) a data into a binary tree
data
data
data
toss a coin
head
private boolean add(TNodeltEgt t, E data) if
(Math.random() lt 0.5) // toss a coin if (t.left
NULL) t.left new TNodeltEgt(data,null,null)
else add(t.left, data) else if (t.right
NULL) t.right new TNodeltEgt(data,null,null)
else add(t.right,data) return true
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Tree traversal preorder, inorder, postorder
X
L
R
X
X
L
R
L
R
inorder L ? X ? R
preorder X ? L ? R
postorder L ? R ? X
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// Three ways of traversals ..... public void
inorder() inorder(root) public void
preorder() preorder(root) public void
postorder() postorder(root) .....
private void inorder(TNodeltEgt t) if
(tNULL) return inorder(t.left)
System.out.print(t.data", ")
inorder(t.right)
// (1) Travel to the left-child // (2) The way we
visit the node // (3) Travel to the right-child
private void preorder(TNodeltEgt t) if
(tNULL) return System.out.print(t.data",
") preorder(t.left)
preorder(t.right)
// (1) The way we visit the node // (2) Travel to
the left-child // (3) Travel to the right-child
private void postorder(TNodeltEgt t) if
(tNULL) return postorder(t.left)
postorder(t.right) System.out.print(t.data
", ")
// (1) Travel to the left-child // (2) Travel to
the right-child // (3) The way we visit the node
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Calculate the size of the tree, i.e., the number
of nodes
X
Ln 1 Rn
root NULL 0
Ln
Rn
// Return the number nodes in the
tree ..... public int size() return size(root)

private int size(TNodeltEgt t) if (tNULL)
return 0 return 1 size(t.left)
size(t.right)
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Count how many ks in the tree
X
root NULL 0
if x k, Ln 1 Rn otherwise, Ln Rn
Ln
Rn
// Return the number of ks in the
tree ..... public int count(E k) return
count(root, k)
private int count(TNodeltEgt t, E k) if
(tNULL) return 0 return (k.compareTo(t.data
) 0 ? 1 0) count(t.left, k)
count(t.right, k)
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Height of a Tree The number of nodes in the
longest path from the root.
height 5
root
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// Return the heights of the tree ..... public
int height() return height(root)
private height(TNodeltTgt t) if (tNULL)
return 0 int l height(t.left) int r
height(t.right) return 1 (l gt r ? l
r)
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Copy and Clone
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root
root
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