Basic Tree

Tree

Definition
Terminology
Property
Storing
Traversal
Binary Tree
Complete Binary Tree

What is a tree ?

Tree - Definition

Definition: non-cyclic connected graph

Wait... what is cyclic? what is connect? graph?

Tree - Definition

Definition: non-cyclic connected graph

Wait... what is cyclic? what is connect? graph?

Cyclic

Connected

non-cyclic

Connected

non-cyclic

not connected

cyclic and not connected is possible

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

0

1

2

3

Tree Terminology

node
edge
root
leaf
parent
child
ancestor
descendant
subtree
level
depth
height

height = 3

height = 2

height = 4

distance to the farest leaf

Tree - Property

Each node can be root

Each node can be root

2. Given any pair of nodes, there exists a unique path that doesn't visit a node twice.

3. A tree have N nodes has exactly (N-1) edges

Tree - Storage

How to store a tree structure

Node is like a linked list !

Question : how many pointers needed?

struct Node{
	int _data;
    Node *_child1, *_child2, *_child3, ...?
    (Node*)* array;
    int index;
}

void addNode(Node * parent, int val){
	if(index == N)
    	array = new Node*[2*N]; 
	array[index] = new Node(val);
    index++;
}
Node::Node(){
	malloc(sizeof(Node*) * N);
}

Tree - Storage

Solution : use linked list or dynamic array (STL vector)

#include <vector>
struct Node{
	int 	_data;
    vector<Node*> _childs;
}

Tree - Storage

Another Solution : Use array of dynamic array (STL vector)

#include <vector>
using namespace std;

int data[SIZE];
vector<int> child[SIZE];

void addChild(int parent, int child){
	child[parent].push_back(child);
}

Use when the Probelm gives the index of each node.

Tree - Traversal

Tree Traversal - Code

#include <vector>
#include <iostream>
using namespace std;
int data[MAXN];
vector<int> child[MAXN];

void DFS_recursive(int root){
	cout << data[root] << " ";
    for(int i = 0; i < child[root].length(); i++){
        DFS_recur(child[root][i]);
    }
}
void DFS_iterative(int root){
	stack<int> tovisited;
    tovisited.push(root);
    while(!tovisited.emtpy())
    {
    	int cur = tovisited.top();
        cout << cur << " ";
        tovisited.pop();
        for(int i = 0; i < child[cur].length(); i++){
        	tovisited.push(child[cur][i]);
        }
    }
}
void BFS(int root){
	queue<int> tovisited;
    tovisited.push(root);
    while(tovisited.empty())
    {
    	int cur = tovisited.front();
        cout << cur << " ";
        tovisited.pop();
        for(int i = 0; i < child[cur].length(); i++){
        	tovisited.push(child[cur][i]);
        }
    }
}

Binary Tree

Each Node has a most two child node
level k has at most nodes
a binary tree with depth k has at most

2^k

2^{k+1}-1 \text{ nodes}

how to traversal ?

pre/ in/ post order

Complete Binary Tree

every level is full except the last level
last level is filled in left first.

Complete Binary Tree

indexs of childs of node k are 2k and 2k+1
parent of node k is [k/2]
depth of a N node complete binary tree is about log(n)

Complete Binary Tree

So we can store it in an array !

1	2	3	4	5	6	7

Why storing a normal binary tree in array is not good ?

Binary Search Tree !

Binary Search Tree - Property

Definition :

if a node has a left subtree, the value of any node on its left subtree is less than the value of itself.
if a node has a right subtree, the value of any node on its right subtree is greater than the value of itself.
left/right subtree of any node is also a binary search tree itself
no duplicate nodes (same value)