Python Data Structures

19 May, 2022
Data Structures are a way of organizing data so that it can be accessed more efficiently depending upon the situation. Data Structures are fundamentals of any programming language around which a program is built. Python helps to learn the fundamental of these data structures in a simpler way as compared to other programming languages.
In this article, we will discuss the Data Structures in the Python Programming Language and how they are related to some specific Python Data Types. We will discuss all the in-built data structures like list tuples, dictionaries, etc. as well as some advanced data structures like trees, graphs, etc.
Python Lists are just like the arrays, declared in other languages which is an ordered collection of data. It is very flexible as the items in a list do not need to be of the same type.
The implementation of Python List is similar to Vectors in C++ or ArrayList in JAVA. The costly operation is inserting or deleting the element from the beginning of the List as all the elements are needed to be shifted. Insertion and deletion at the end of the list can also become costly in the case where the preallocated memory becomes full.
We can create a list in python as shown below.
Python3
List = [1, 2,  3, "GFG", 2.3]print(List)
[1, 2, 3, 'GFG', 2.3]
List elements can be accessed by the assigned index. In python starting index of the list, sequence is 0 and the ending index is (if N elements are there) N-1.
Python3
# Creating a List with # the use of multiple values List = ["Geeks", "For", "Geeks"] print("\nList containing multiple values: ") print(List)  # Creating a Multi-Dimensional List # (By Nesting a list inside a List) List2 = [['Geeks', 'For'], ['Geeks']] print("\nMulti-Dimensional List: ") print(List2)   # accessing a element from the  # list using index number print("Accessing element from the list") print(List[0])  print(List[2])   # accessing a element using # negative indexing print("Accessing element using negative indexing")       # print the last element of list print(List[-1])       # print the third last element of list  print(List[-3])
List containing multiple values: 
['Geeks', 'For', 'Geeks']

Multi-Dimensional List: 
[['Geeks', 'For'], ['Geeks']]
Accessing element from the list
Geeks
Geeks
Accessing element using negative indexing
Geeks
Geeks
Python dictionary is like hash tables in any other language with the time complexity of O(1). It is an unordered collection of data values, used to store data values like a map, which, unlike other Data Types that hold only a single value as an element, Dictionary holds the key:value pair. Key-value is provided in the dictionary to make it more optimized. 
Indexing of Python Dictionary is done with the help of keys. These are of any hashable type i.e. an object whose can never change like strings, numbers, tuples, etc. We can create a dictionary by using curly braces ({}) or dictionary comprehension.
Python3
# Creating a DictionaryDict = {'Name': 'Geeks', 1: [1, 2, 3, 4]}print("Creating Dictionary: ")print(Dict)  # accessing a element using key print("Accessing a element using key:") print(Dict['Name'])   # accessing a element using get() # method print("Accessing a element using get:") print(Dict.get(1))   # creation using Dictionary comprehensionmyDict = {x: x**2 for x in [1,2,3,4,5]}print(myDict)
Creating Dictionary: 
{'Name': 'Geeks', 1: [1, 2, 3, 4]}
Accessing a element using key:
Geeks
Accessing a element using get:
[1, 2, 3, 4]
{1: 1, 2: 4, 3: 9, 4: 16, 5: 25}
Python Tuple is a collection of Python objects much like a list but Tuples are immutable in nature i.e. the elements in the tuple cannot be added or removed once created. Just like a List, a Tuple can also contain elements of various types.
In Python, tuples are created by placing a sequence of values separated by ‘comma’ with or without the use of parentheses for grouping of the data sequence.
Note: Tuples can also be created with a single element, but it is a bit tricky. Having one element in the parentheses is not sufficient, there must be a trailing ‘comma’ to make it a tuple.
Python3
# Creating a Tuple with# the use of StringsTuple = ('Geeks', 'For')print("\nTuple with the use of String: ")print(Tuple)      # Creating a Tuple with# the use of listlist1 = [1, 2, 4, 5, 6]print("\nTuple using List: ")Tuple = tuple(list1)  # Accessing element using indexingprint("First element of tuple")print(Tuple[0])  # Accessing element from last# negative indexingprint("\nLast element of tuple")print(Tuple[-1])    print("\nThird last element of tuple")print(Tuple[-3])
Tuple with the use of String: 
('Geeks', 'For')

Tuple using List: 
First element of tuple
1

Last element of tuple
6

Third last element of tuple
4
Python Set is an ordered collection of data that is mutable and does not allow any duplicate element. Sets are basically used to include membership testing and eliminating duplicate entries. The data structure used in this is Hashing, a popular technique to perform insertion, deletion, and traversal in O(1) on average. 
If Multiple values are present at the same index position, then the value is appended to that index position, to form a Linked List. In, CPython Sets are implemented using a dictionary with dummy variables, where key beings the members set with greater optimizations to the time complexity.
Set Implementation:
Sets with Numerous operations on a single HashTable:
Python3
# Creating a Set with  # a mixed type of values # (Having numbers and strings) Set = set([1, 2, 'Geeks', 4, 'For', 6, 'Geeks']) print("\nSet with the use of Mixed Values") print(Set)   # Accessing element using # for loop print("\nElements of set: ") for i in Set:     print(i, end =" ") print()  # Checking the element # using in keyword print("Geeks" in Set)
Set with the use of Mixed Values
{1, 2, 'Geeks', 4, 6, 'For'}

Elements of set: 
1 2 Geeks 4 6 For 
True
Frozen sets in Python are immutable objects that only support methods and operators that produce a result without affecting the frozen set or sets to which they are applied. While elements of a set can be modified at any time, elements of the frozen set remain the same after creation.
If no parameters are passed, it returns an empty frozenset.
Python3
# Same as {"a", "b","c"}normal_set = set(["a", "b","c"])  print("Normal Set")print(normal_set)  # A frozen setfrozen_set = frozenset(["e", "f", "g"])  print("\nFrozen Set")print(frozen_set)  # Uncommenting below line would cause error as# we are trying to add element to a frozen set# frozen_set.add("h")
Normal Set
{'a', 'c', 'b'}

Frozen Set
frozenset({'g', 'e', 'f'})
Python Strings are arrays of bytes representing Unicode characters. In simpler terms, a string is an immutable array of characters. Python does not have a character data type, a single character is simply a string with a length of 1.
Note: As strings are immutable, modifying a string will result in creating a new copy.
Python3
String = "Welcome to GeeksForGeeks"print("Creating String: ") print(String)       # Printing First character print("\nFirst character of String is: ") print(String[0])       # Printing Last character print("\nLast character of String is: ") print(String[-1])
Creating String: 
Welcome to GeeksForGeeks

First character of String is: 
W

Last character of String is: 
s
Python Bytearray gives a mutable sequence of integers in the range 0 <= x < 256.
Python3
# Creating bytearraya = bytearray((12, 8, 25, 2))print("Creating Bytearray:")print(a)  # accessing elementsprint("\nAccessing Elements:", a[1])  # modifying elements a[1] = 3print("\nAfter Modifying:")print(a)  # Appending elementsa.append(30)print("\nAfter Adding Elements:")print(a)
Creating Bytearray:
bytearray(b'\x0c\x08\x19\x02')

Accessing Elements: 8

After Modifying:
bytearray(b'\x0c\x03\x19\x02')

After Adding Elements:
bytearray(b'\x0c\x03\x19\x02\x1e')
Till now we have studied all the data structures that come built-in into core Python. Now let dive more deep into Python and see the collections module that provides some containers that are useful in many cases and provide more features than the above-defined functions.
Python collection module was introduced to improve the functionality of the built-in datatypes. It provides various containers let’s see each one of them in detail.
A counter is a sub-class of the dictionary. It is used to keep the count of the elements in an iterable in the form of an unordered dictionary where the key represents the element in the iterable and value represents the count of that element in the iterable. This is equivalent to a bag or multiset of other languages.
Python3
from collections import Counter    # With sequence of items print(Counter(['B','B','A','B','C','A','B','B','A','C']))    # with dictionarycount = Counter({'A':3, 'B':5, 'C':2})print(count)  count.update(['A', 1])print(count)
Counter({'B': 5, 'A': 3, 'C': 2})
Counter({'B': 5, 'A': 3, 'C': 2})
Counter({'B': 5, 'A': 4, 'C': 2, 1: 1})
An OrderedDict is also a sub-class of dictionary but unlike a dictionary, it remembers the order in which the keys were inserted. 
Python3
from collections import OrderedDict  print("Before deleting:\n")od = OrderedDict()od['a'] = 1od['b'] = 2od['c'] = 3od['d'] = 4  for key, value in od.items():    print(key, value)  print("\nAfter deleting:\n")od.pop('c')for key, value in od.items():    print(key, value)  print("\nAfter re-inserting:\n")od['c'] = 3for key, value in od.items():    print(key, value)
Before deleting:

a 1
b 2
c 3
d 4

After deleting:

a 1
b 2
d 4

After re-inserting:

a 1
b 2
d 4
c 3
DefaultDict is used to provide some default values for the key that does not exist and never raises a KeyError. Its objects can be initialized using DefaultDict() method by passing the data type as an argument.
Note: default_factory is a function that provides the default value for the dictionary created. If this parameter is absent then the KeyError is raised.
Python3
from collections import defaultdict        # Defining the dictd = defaultdict(int)      L = [1, 2, 3, 4, 2, 4, 1, 2]      # Iterate through the list# for keeping the countfor i in L:              # The default value is 0    # so there is no need to    # enter the key first    d[i] += 1          print(d)
defaultdict(<class 'int'>, {1: 2, 2: 3, 3: 1, 4: 2})
A ChainMap encapsulates many dictionaries into a single unit and returns a list of dictionaries. When a key is needed to be found then all the dictionaries are searched one by one until the key is found.
Python3
from collections import ChainMap            d1 = {'a': 1, 'b': 2}d2 = {'c': 3, 'd': 4}d3 = {'e': 5, 'f': 6}      # Defining the chainmapc = ChainMap(d1, d2, d3)print(c)  print(c['a'])print(c['g'])
Output
ChainMap({'a': 1, 'b': 2}, {'c': 3, 'd': 4}, {'e': 5, 'f': 6})
1
KeyError: 'g'
A NamedTuple returns a tuple object with names for each position which the ordinary tuples lack. For example, consider a tuple names student where the first element represents fname, second represents lname and the third element represents the DOB. Suppose for calling fname instead of remembering the index position you can actually call the element by using the fname argument, then it will be really easy for accessing tuples element. This functionality is provided by the NamedTuple.
Python3
from collections import namedtuple      # Declaring namedtuple()Student = namedtuple('Student',['name','age','DOB'])      # Adding valuesS = Student('Nandini','19','2541997')      # Access using indexprint ("The Student age using index is : ",end ="")print (S[1])      # Access using nameprint ("The Student name using keyname is : ",end ="")print (S.name)
The Student age using index is : 19
The Student name using keyname is : Nandini
Deque (Doubly Ended Queue) is the optimized list for quicker append and pop operations from both sides of the container. It provides O(1) time complexity for append and pop operations as compared to the list with O(n) time complexity.
Python Deque is implemented using doubly linked lists therefore the performance for randomly accessing the elements is O(n).
Python3
# importing "collections" for deque operationsimport collections  # initializing dequede = collections.deque([1,2,3])  # using append() to insert element at right end# inserts 4 at the end of dequede.append(4)  # printing modified dequeprint("The deque after appending at right is : ")print(de)  # using appendleft() to insert element at left end# inserts 6 at the beginning of dequede.appendleft(6)  # printing modified dequeprint("The deque after appending at left is : ")print(de)  # using pop() to delete element from right end# deletes 4 from the right end of dequede.pop()  # printing modified dequeprint("The deque after deleting from right is : ")print(de)  # using popleft() to delete element from left end# deletes 6 from the left end of dequede.popleft()  # printing modified dequeprint("The deque after deleting from left is : ")print(de)
The deque after appending at right is : 
deque([1, 2, 3, 4])
The deque after appending at left is : 
deque([6, 1, 2, 3, 4])
The deque after deleting from right is : 
deque([6, 1, 2, 3])
The deque after deleting from left is : 
deque([1, 2, 3])
UserDict is a dictionary-like container that acts as a wrapper around the dictionary objects. This container is used when someone wants to create their own dictionary with some modified or new functionality. 
Python3
from collections import UserDict  # Creating a Dictionary where# deletion is not allowedclass MyDict(UserDict):          # Function to stop deletion    # from dictionary    def __del__(self):        raise RuntimeError("Deletion not allowed")              # Function to stop pop from    # dictionary    def pop(self, s = None):        raise RuntimeError("Deletion not allowed")              # Function to stop popitem    # from Dictionary    def popitem(self, s = None):        raise RuntimeError("Deletion not allowed")      # Driver's coded = MyDict({'a':1,    'b': 2,    'c': 3})  print("Original Dictionary")print(d)  d.pop(1)
Output
Original Dictionary
{'a': 1, 'b': 2, 'c': 3}
RuntimeError: Deletion not allowed
UserList is a list-like container that acts as a wrapper around the list objects. This is useful when someone wants to create their own list with some modified or additional functionality.
Python3
# Python program to demonstrate# userlist    from collections import UserList    # Creating a List where# deletion is not allowedclass MyList(UserList):          # Function to stop deletion    # from List    def remove(self, s = None):        raise RuntimeError("Deletion not allowed")              # Function to stop pop from    # List    def pop(self, s = None):        raise RuntimeError("Deletion not allowed")      # Driver's codeL = MyList([1, 2, 3, 4])  print("Original List")print(L)  # Inserting to List"L.append(5)print("After Insertion")print(L)  # Deleting From ListL.remove()
Output
Original List
[1, 2, 3, 4]
After Insertion
[1, 2, 3, 4, 5]
RuntimeError: Deletion not allowed
UserString is a string-like container and just like UserDict and UserList, it acts as a wrapper around string objects. It is used when someone wants to create their own strings with some modified or additional functionality. 
Python3
from collections import UserString  # Creating a Mutable Stringclass Mystring(UserString):          # Function to append to    # string    def append(self, s):        self.data += s              # Function to remove from    # string    def remove(self, s):        self.data = self.data.replace(s, "")      # Driver's codes1 = Mystring("Geeks")print("Original String:", s1.data)  # Appending to strings1.append("s")print("String After Appending:", s1.data)  # Removing from strings1.remove("e")print("String after Removing:", s1.data)
Original String: Geeks
String After Appending: Geekss
String after Removing: Gkss
Now after studying all the data structures let’s see some advanced data structures such as stack, queue, graph, linked list, etc. that can be used in Python Language.
A linked list is a linear data structure, in which the elements are not stored at contiguous memory locations. The elements in a linked list are linked using pointers as shown in the below image:
A linked list is represented by a pointer to the first node of the linked list. The first node is called the head. If the linked list is empty, then the value of the head is NULL. Each node in a list consists of at least two parts:
Data
Pointer (Or Reference) to the next node
Python3
# Node classclass Node:      # Function to initialize the node object    def __init__(self, data):        self.data = data # Assign data        self.next = None # Initialize                        # next as null  # Linked List classclass LinkedList:          # Function to initialize the Linked    # List object    def __init__(self):        self.head = None
Let us create a simple linked list with 3 nodes.
Python3
# A simple Python program to introduce a linked list  # Node classclass Node:      # Function to initialise the node object    def __init__(self, data):        self.data = data # Assign data        self.next = None # Initialize next as null    # Linked List class contains a Node objectclass LinkedList:      # Function to initialize head    def __init__(self):        self.head = None    # Code execution starts hereif __name__=='__main__':      # Start with the empty list    llist = LinkedList()      llist.head = Node(1)    second = Node(2)    third = Node(3)      '''    Three nodes have been created.    We have references to these three blocks as head,    second and third      llist.head     second             third        |             |                 |        |             |                 |    +----+------+     +----+------+     +----+------+    | 1 | None |     | 2 | None |     | 3 | None |    +----+------+     +----+------+     +----+------+    '''      llist.head.next = second; # Link first node with second      '''    Now next of first Node refers to second. So they    both are linked.      llist.head     second             third        |             |                 |        |             |                 |    +----+------+     +----+------+     +----+------+    | 1 | o-------->| 2 | null |     | 3 | null |    +----+------+     +----+------+     +----+------+    '''      second.next = third; # Link second node with the third node      '''    Now next of second Node refers to third. So all three    nodes are linked.      llist.head     second             third        |             |                 |        |             |                 |    +----+------+     +----+------+     +----+------+    | 1 | o-------->| 2 | o-------->| 3 | null |    +----+------+     +----+------+     +----+------+    '''
In the previous program, we have created a simple linked list with three nodes. Let us traverse the created list and print the data of each node. For traversal, let us write a general-purpose function printList() that prints any given list.
Python3
# A simple Python program for traversal of a linked list  # Node classclass Node:      # Function to initialise the node object    def __init__(self, data):        self.data = data # Assign data        self.next = None # Initialize next as null    # Linked List class contains a Node objectclass LinkedList:      # Function to initialize head    def __init__(self):        self.head = None      # This function prints contents of linked list    # starting from head    def printList(self):        temp = self.head        while (temp):            print (temp.data)            temp = temp.next    # Code execution starts hereif __name__=='__main__':      # Start with the empty list    llist = LinkedList()      llist.head = Node(1)    second = Node(2)    third = Node(3)      llist.head.next = second; # Link first node with second    second.next = third; # Link second node with the third node      llist.printList()
1
2
3
A stack is a linear data structure that stores items in a Last-In/First-Out (LIFO) or First-In/Last-Out (FILO) manner. In stack, a new element is added at one end and an element is removed from that end only. The insert and delete operations are often called push and pop.
The functions associated with stack are:
empty() – Returns whether the stack is empty – Time Complexity: O(1)
size() – Returns the size of the stack – Time Complexity: O(1)
top() – Returns a reference to the topmost element of the stack – Time Complexity: O(1)
push(a) – Inserts the element ‘a’ at the top of the stack – Time Complexity: O(1)
pop() – Deletes the topmost element of the stack – Time Complexity: O(1)
Stack in Python can be implemented using the following ways:
list
Collections.deque
queue.LifoQueue
Implementation using List
Python’s built-in data structure list can be used as a stack. Instead of push(), append() is used to add elements to the top of the stack while pop() removes the element in LIFO order. 
Python3
stack = []  # append() function to push# element in the stackstack.append('g')stack.append('f')stack.append('g')  print('Initial stack')print(stack)  # pop() function to pop# element from stack in# LIFO orderprint('\nElements popped from stack:')print(stack.pop())print(stack.pop())print(stack.pop())  print('\nStack after elements are popped:')print(stack)  # uncommenting print(stack.pop())# will cause an IndexError# as the stack is now empty
Initial stack
['g', 'f', 'g']

Elements popped from stack:
g
f
g

Stack after elements are popped:
[]
Implementation using collections.deque:
Python stack can be implemented using the deque class from the collections module. Deque is preferred over the list in the cases where we need quicker append and pop operations from both the ends of the container, as deque provides an O(1) time complexity for append and pop operations as compared to list which provides O(n) time complexity. 
Python3
from collections import deque  stack = deque()  # append() function to push# element in the stackstack.append('g')stack.append('f')stack.append('g')  print('Initial stack:')print(stack)  # pop() function to pop# element from stack in# LIFO orderprint('\nElements popped from stack:')print(stack.pop())print(stack.pop())print(stack.pop())  print('\nStack after elements are popped:')print(stack)  # uncommenting print(stack.pop())# will cause an IndexError# as the stack is now empty
Initial stack:
deque(['g', 'f', 'g'])

Elements popped from stack:
g
f
g

Stack after elements are popped:
deque([])
Implementation using queue module
The queue module also has a LIFO Queue, which is basically a Stack. Data is inserted into Queue using the put() function and get() takes data out from the Queue. 
Python3
from queue import LifoQueue  # Initializing a stackstack = LifoQueue(maxsize = 3)  # qsize() show the number of elements# in the stackprint(stack.qsize())  # put() function to push# element in the stackstack.put('g')stack.put('f')stack.put('g')  print("Full: ", stack.full())print("Size: ", stack.qsize())  # get() function to pop# element from stack in# LIFO orderprint('\nElements popped from the stack')print(stack.get())print(stack.get())print(stack.get())  print("\nEmpty: ", stack.empty())
0
Full:  True
Size:  3

Elements popped from the stack
g
f
g

Empty:  True
As a stack, the queue is a linear data structure that stores items in a First In First Out (FIFO) manner. With a queue, the least recently added item is removed first. A good example of the queue is any queue of consumers for a resource where the consumer that came first is served first.
Operations associated with queue are:
Enqueue: Adds an item to the queue. If the queue is full, then it is said to be an Overflow condition – Time Complexity: O(1)
Dequeue: Removes an item from the queue. The items are popped in the same order in which they are pushed. If the queue is empty, then it is said to be an Underflow condition – Time Complexity: O(1)
Front: Get the front item from queue – Time Complexity: O(1)
Rear: Get the last item from queue – Time Complexity: O(1)
Queue in Python can be implemented in the following ways:
list
collections.deque
queue.Queue
Implementation using list
Instead of enqueue() and dequeue(), append() and pop() function is used.
Python3
# Initializing a queuequeue = []  # Adding elements to the queuequeue.append('g')queue.append('f')queue.append('g')  print("Initial queue")print(queue)  # Removing elements from the queueprint("\nElements dequeued from queue")print(queue.pop(0))print(queue.pop(0))print(queue.pop(0))  print("\nQueue after removing elements")print(queue)  # Uncommenting print(queue.pop(0))# will raise and IndexError# as the queue is now empty
Initial queue
['g', 'f', 'g']

Elements dequeued from queue
g
f
g

Queue after removing elements
[]
Implementation using collections.deque
Deque is preferred over the list in the cases where we need quicker append and pop operations from both the ends of the container, as deque provides an O(1) time complexity for append and pop operations as compared to list which provides O(n) time complexity.
Python3
from collections import deque  # Initializing a queueq = deque()  # Adding elements to a queueq.append('g')q.append('f')q.append('g')  print("Initial queue")print(q)  # Removing elements from a queueprint("\nElements dequeued from the queue")print(q.popleft())print(q.popleft())print(q.popleft())  print("\nQueue after removing elements")print(q)  # Uncommenting q.popleft()# will raise an IndexError# as queue is now empty
Initial queue
deque(['g', 'f', 'g'])

Elements dequeued from the queue
g
f
g

Queue after removing elements
deque([])
Implementation using the queue.Queue
queue.Queue(maxsize) initializes a variable to a maximum size of maxsize. A maxsize of zero ‘0’ means an infinite queue. This Queue follows the FIFO rule. 
Python3
from queue import Queue  # Initializing a queueq = Queue(maxsize = 3)  # qsize() give the maxsize# of the Queueprint(q.qsize())  # Adding of element to queueq.put('g')q.put('f')q.put('g')  # Return Boolean for Full# Queueprint("\nFull: ", q.full())  # Removing element from queueprint("\nElements dequeued from the queue")print(q.get())print(q.get())print(q.get())  # Return Boolean for Empty# Queueprint("\nEmpty: ", q.empty())  q.put(1)print("\nEmpty: ", q.empty())print("Full: ", q.full())  # This would result into Infinite# Loop as the Queue is empty.# print(q.get())
0

Full:  True

Elements dequeued from the queue
g
f
g

Empty:  True

Empty:  False
Full:  False
Priority Queues are abstract data structures where each data/value in the queue has a certain priority. For example, In airlines, baggage with the title “Business” or “First-class” arrives earlier than the rest. Priority Queue is an extension of the queue with the following properties.
An element with high priority is dequeued before an element with low priority.
If two elements have the same priority, they are served according to their order in the queue.
Python3
# A simple implementation of Priority Queue# using Queue.class PriorityQueue(object):    def __init__(self):        self.queue = []      def __str__(self):        return ' '.join([str(i) for i in self.queue])      # for checking if the queue is empty    def isEmpty(self):        return len(self.queue) == 0      # for inserting an element in the queue    def insert(self, data):        self.queue.append(data)      # for popping an element based on Priority    def delete(self):        try:            max = 0            for i in range(len(self.queue)):                if self.queue[i] > self.queue[max]:                    max = i            item = self.queue[max]            del self.queue[max]            return item        except IndexError:            print()            exit()  if __name__ == '__main__':    myQueue = PriorityQueue()    myQueue.insert(12)    myQueue.insert(1)    myQueue.insert(14)    myQueue.insert(7)    print(myQueue)                while not myQueue.isEmpty():        print(myQueue.delete())
12 1 14 7
14
12
7
1
heapq module in Python provides the heap data structure that is mainly used to represent a priority queue. The property of this data structure in Python is that each time the smallest heap element is popped(min-heap). Whenever elements are pushed or popped, heap structure is maintained. The heap[0] element also returns the smallest element each time. 
It supports the extraction and insertion of the smallest element in the O(log n) times.
Python3
# importing "heapq" to implement heap queueimport heapq  # initializing listli = [5, 7, 9, 1, 3]  # using heapify to convert list into heapheapq.heapify(li)  # printing created heapprint ("The created heap is : ",end="")print (list(li))  # using heappush() to push elements into heap# pushes 4heapq.heappush(li,4)  # printing modified heapprint ("The modified heap after push is : ",end="")print (list(li))  # using heappop() to pop smallest elementprint ("The popped and smallest element is : ",end="")print (heapq.heappop(li))
The created heap is : [1, 3, 9, 7, 5]
The modified heap after push is : [1, 3, 4, 7, 5, 9]
The popped and smallest element is : 1
A tree is a  hierarchical data structure that looks like the below figure – 
      tree
     ----
      j    <-- root
    /   \
   f      k  
 /   \      \
a     h      z    <-- leaves
The topmost node of the tree is called the root whereas the bottommost nodes or the nodes with no children are called the leaf nodes. The nodes that are directly under a node are called its children and the nodes that are directly above something are called its parent.
A binary tree is a tree whose elements can have almost two children. Since each element in a binary tree can have only 2 children, we typically name them the left and right children. A Binary Tree node contains the following parts.
Data
Pointer to left child
Pointer to the right child
Python3
# A Python class that represents an individual node# in a Binary Treeclass Node:    def __init__(self,key):        self.left = None        self.right = None        self.val = key
Now let’s create a tree with 4 nodes in Python. Let’s assume the tree structure looks like below – 
      tree
     ----
      1    <-- root
    /   \
   2     3  
  /  
 4
Python3
# Python program to introduce Binary Tree  # A class that represents an individual node in a# Binary Treeclass Node:    def __init__(self,key):        self.left = None        self.right = None        self.val = key    # create rootroot = Node(1)''' following is the tree after above statement        1    / \    None None'''  root.left     = Node(2);root.right     = Node(3);  ''' 2 and 3 become left and right children of 1        1        / \        2     3    / \ / \None None None None'''    root.left.left = Node(4);'''4 becomes left child of 2        1    /     \    2         3    / \     / \4 None None None/ \None None'''
Trees can be traversed in different ways. Following are the generally used ways for traversing trees. Let us consider the below tree – 
      tree
     ----
      1    <-- root
    /   \
   2     3  
  / \
 4   5
Depth First Traversals:
Inorder (Left, Root, Right) : 4 2 5 1 3
Preorder (Root, Left, Right) : 1 2 4 5 3
Postorder (Left, Right, Root) : 4 5 2 3 1
Algorithm Inorder(tree)
Traverse the left subtree, i.e., call Inorder(left-subtree)
Visit the root.
Traverse the right subtree, i.e., call Inorder(right-subtree)
Algorithm Preorder(tree)
Visit the root.
Traverse the left subtree, i.e., call Preorder(left-subtree)
Traverse the right subtree, i.e., call Preorder(right-subtree)
Algorithm Postorder(tree)
Traverse the left subtree, i.e., call Postorder(left-subtree)
Traverse the right subtree, i.e., call Postorder(right-subtree)
Visit the root.
Python3
# Python program to for tree traversals  # A class that represents an individual node in a# Binary Treeclass Node:    def __init__(self, key):        self.left = None        self.right = None        self.val = key    # A function to do inorder tree traversaldef printInorder(root):      if root:          # First recur on left child        printInorder(root.left)          # then print the data of node        print(root.val),          # now recur on right child        printInorder(root.right)    # A function to do postorder tree traversaldef printPostorder(root):      if root:          # First recur on left child        printPostorder(root.left)          # the recur on right child        printPostorder(root.right)          # now print the data of node        print(root.val),    # A function to do preorder tree traversaldef printPreorder(root):      if root:          # First print the data of node        print(root.val),          # Then recur on left child        printPreorder(root.left)          # Finally recur on right child        printPreorder(root.right)    # Driver coderoot = Node(1)root.left = Node(2)root.right = Node(3)root.left.left = Node(4)root.left.right = Node(5)print("Preorder traversal of binary tree is")printPreorder(root)  print("\nInorder traversal of binary tree is")printInorder(root)  print("\nPostorder traversal of binary tree is")printPostorder(root)
Preorder traversal of binary tree is
1
2
4
5
3

Inorder traversal of binary tree is
4
2
5
1
3

Postorder traversal of binary tree is
4
5
2
3
1
Time Complexity –  O(n) 
Breadth-First or Level Order Traversal
Level order traversal of a tree is breadth-first traversal for the tree. The level order traversal of the above tree is 1 2 3 4 5.
For each node, first, the node is visited and then its child nodes are put in a FIFO queue. Below is the algorithm for the same – 
Create an empty queue q
temp_node = root /*start from root*/
Loop while temp_node is not NULLprint temp_node->data.Enqueue temp_node’s children (first left then right children) to qDequeue a node from q
print temp_node->data.
Enqueue temp_node’s children (first left then right children) to q
Dequeue a node from q
Python3
# Python program to print level# order traversal using Queue  # A node structureclass Node:        # A utility function to create a new node    def __init__(self ,key):        self.data = key        self.left = None        self.right = None  # Iterative Method to print the# height of a binary treedef printLevelOrder(root):        # Base Case    if root is None:        return          # Create an empty queue    # for level order traversal    queue = []      # Enqueue Root and initialize height    queue.append(root)      while(len(queue) > 0):              # Print front of queue and        # remove it from queue        print (queue[0].data)        node = queue.pop(0)          # Enqueue left child        if node.left is not None:            queue.append(node.left)          # Enqueue right child        if node.right is not None:            queue.append(node.right)  # Driver Program to test above functionroot = Node(1)root.left = Node(2)root.right = Node(3)root.left.left = Node(4)root.left.right = Node(5)  print ("Level Order Traversal of binary tree is -")printLevelOrder(root)
Level Order Traversal of binary tree is -
1
2
3
4
5
Time Complexity: O(n) 
A graph is a nonlinear data structure consisting of nodes and edges. The nodes are sometimes also referred to as vertices and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph can be defined as a Graph consisting of a finite set of vertices(or nodes) and a set of edges that connect a pair of nodes.
In the above Graph, the set of vertices V = {0,1,2,3,4} and the set of edges E = {01, 12, 23, 34, 04, 14, 13}.
The following two are the most commonly used representations of a graph.
Adjacency Matrix
Adjacency List
Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. The adjacency matrix for an undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. 
Python3
# A simple representation of graph using Adjacency Matrixclass Graph:    def __init__(self,numvertex):        self.adjMatrix = [[-1]*numvertex for x in range(numvertex)]        self.numvertex = numvertex        self.vertices = {}        self.verticeslist =[0]*numvertex      def set_vertex(self,vtx,id):        if 0<=vtx<=self.numvertex:            self.vertices[id] = vtx            self.verticeslist[vtx] = id      def set_edge(self,frm,to,cost=0):        frm = self.vertices[frm]        to = self.vertices[to]        self.adjMatrix[frm][to] = cost                  # for directed graph do not add this        self.adjMatrix[to][frm] = cost      def get_vertex(self):        return self.verticeslist      def get_edges(self):        edges=[]        for i in range (self.numvertex):            for j in range (self.numvertex):                if (self.adjMatrix[i][j]!=-1):                    edges.append((self.verticeslist[i],self.verticeslist[j],self.adjMatrix[i][j]))        return edges              def get_matrix(self):        return self.adjMatrix  G =Graph(6)G.set_vertex(0,'a')G.set_vertex(1,'b')G.set_vertex(2,'c')G.set_vertex(3,'d')G.set_vertex(4,'e')G.set_vertex(5,'f')G.set_edge('a','e',10)G.set_edge('a','c',20)G.set_edge('c','b',30)G.set_edge('b','e',40)G.set_edge('e','d',50)G.set_edge('f','e',60)  print("Vertices of Graph")print(G.get_vertex())  print("Edges of Graph")print(G.get_edges())  print("Adjacency Matrix of Graph")print(G.get_matrix())
Output
Vertices of Graph
[‘a’, ‘b’, ‘c’, ‘d’, ‘e’, ‘f’]
Edges of Graph
[(‘a’, ‘c’, 20), (‘a’, ‘e’, 10), (‘b’, ‘c’, 30), (‘b’, ‘e’, 40), (‘c’, ‘a’, 20), (‘c’, ‘b’, 30), (‘d’, ‘e’, 50), (‘e’, ‘a’, 10), (‘e’, ‘b’, 40), (‘e’, ‘d’, 50), (‘e’, ‘f’, 60), (‘f’, ‘e’, 60)]
Adjacency Matrix of Graph
[[-1, -1, 20, -1, 10, -1], [-1, -1, 30, -1, 40, -1], [20, 30, -1, -1, -1, -1], [-1, -1, -1, -1, 50, -1], [10, 40, -1, 50, -1, 60], [-1, -1, -1, -1, 60, -1]]
An array of lists is used. The size of the array is equal to the number of vertices. Let the array be an array[]. An entry array[i] represents the list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is the adjacency list representation of the above graph. 
 
Python3
# A class to represent the adjacency list of the nodeclass AdjNode:    def __init__(self, data):        self.vertex = data        self.next = None    # A class to represent a graph. A graph# is the list of the adjacency lists.# Size of the array will be the no. of the# vertices "V"class Graph:    def __init__(self, vertices):        self.V = vertices        self.graph = [None] * self.V      # Function to add an edge in an undirected graph    def add_edge(self, src, dest):                # Adding the node to the source node        node = AdjNode(dest)        node.next = self.graph[src]        self.graph[src] = node          # Adding the source node to the destination as        # it is the undirected graph        node = AdjNode(src)        node.next = self.graph[dest]        self.graph[dest] = node      # Function to print the graph    def print_graph(self):        for i in range(self.V):            print("Adjacency list of vertex {}\n head".format(i), end="")            temp = self.graph[i]            while temp:                print(" -> {}".format(temp.vertex), end="")                temp = temp.next            print(" \n")    # Driver program to the above graph classif __name__ == "__main__":    V = 5    graph = Graph(V)    graph.add_edge(0, 1)    graph.add_edge(0, 4)    graph.add_edge(1, 2)    graph.add_edge(1, 3)    graph.add_edge(1, 4)    graph.add_edge(2, 3)    graph.add_edge(3, 4)      graph.print_graph()
Adjacency list of vertex 0
 head -> 4 -> 1 

Adjacency list of vertex 1
 head -> 4 -> 3 -> 2 -> 0 

Adjacency list of vertex 2
 head -> 3 -> 1 

Adjacency list of vertex 3
 head -> 4 -> 2 -> 1 

Adjacency list of vertex 4
 head -> 3 -> 1 -> 0 
Breadth-First Search or BFS
Breadth-First Traversal for a graph is similar to Breadth-First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles, so we may come to the same node again. To avoid processing a node more than once, we use a boolean visited array. For simplicity, it is assumed that all vertices are reachable from the starting vertex.
For example, in the following graph, we start traversal from vertex 2. When we come to vertex 0, we look for all adjacent vertices of it. 2 is also an adjacent vertex of 0. If we don’t mark visited vertices, then 2 will be processed again and it will become a non-terminating process. A Breadth-First Traversal of the following graph is 2, 0, 3, 1.
 
Python3
# Python3 Program to print BFS traversal# from a given source vertex. BFS(int s)# traverses vertices reachable from s.from collections import defaultdict  # This class represents a directed graph# using adjacency list representationclass Graph:      # Constructor    def __init__(self):          # default dictionary to store graph        self.graph = defaultdict(list)      # function to add an edge to graph    def addEdge(self,u,v):        self.graph[u].append(v)      # Function to print a BFS of graph    def BFS(self, s):          # Mark all the vertices as not visited        visited = [False] * (max(self.graph) + 1)          # Create a queue for BFS        queue = []          # Mark the source node as        # visited and enqueue it        queue.append(s)        visited[s] = True          while queue:              # Dequeue a vertex from            # queue and print it            s = queue.pop(0)            print (s, end = " ")              # Get all adjacent vertices of the            # dequeued vertex s. If a adjacent            # has not been visited, then mark it            # visited and enqueue it            for i in self.graph[s]:                if visited[i] == False:                    queue.append(i)                    visited[i] = True  # Driver code  # Create a graph given in# the above diagramg = Graph()g.addEdge(0, 1)g.addEdge(0, 2)g.addEdge(1, 2)g.addEdge(2, 0)g.addEdge(2, 3)g.addEdge(3, 3)  print ("Following is Breadth First Traversal"                " (starting from vertex 2)")g.BFS(2)  # This code is contributed by Neelam Yadav
Following is Breadth First Traversal (starting from vertex 2)
2 0 3 1 
Time Complexity: O(V+E) where V is the number of vertices in the graph and E is the number of edges in the graph.
Depth First Search or DFS
Depth First Traversal for a graph is similar to Depth First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles, a node may be visited twice. To avoid processing a node more than once, use a boolean visited array.
Algorithm:
Create a recursive function that takes the index of the node and a visited array.
Mark the current node as visited and print the node.
Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.
Python3
# Python3 program to print DFS traversal# from a given graphfrom collections import defaultdict  # This class represents a directed graph using# adjacency list representation    class Graph:      # Constructor    def __init__(self):          # default dictionary to store graph        self.graph = defaultdict(list)      # function to add an edge to graph    def addEdge(self, u, v):        self.graph[u].append(v)      # A function used by DFS    def DFSUtil(self, v, visited):          # Mark the current node as visited        # and print it        visited.add(v)        print(v, end=' ')          # Recur for all the vertices        # adjacent to this vertex        for neighbour in self.graph[v]:            if neighbour not in visited:                self.DFSUtil(neighbour, visited)      # The function to do DFS traversal. It uses    # recursive DFSUtil()    def DFS(self, v):          # Create a set to store visited vertices        visited = set()          # Call the recursive helper function        # to print DFS traversal        self.DFSUtil(v, visited)  # Driver code    # Create a graph given# in the above diagramg = Graph()g.addEdge(0, 1)g.addEdge(0, 2)g.addEdge(1, 2)g.addEdge(2, 0)g.addEdge(2, 3)g.addEdge(3, 3)  print("Following is DFS from (starting from vertex 2)")g.DFS(2)
Following is DFS from (starting from vertex 2)
2 0 1 3 
Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
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