1. A leftist heap is a type of priority queue that is implemented using a variant of a binary heap. It attempts to be very unbalanced in contrast to a binary heap, which is always a complete binary tree. 2. Key operations on a leftist heap like insert, delete minimum, and merge have O(logN) time complexity. Merge is particularly efficient compared to a binary heap which has O(N) time complexity for merging. 3. Properties of a leftist heap include the minimum key property and the property that the right descendant of every node has a smaller s-value (distance to the nearest leaf). These properties allow leftist heaps to efficiently support merge in O(log

1. Introduction
A leftist heap or leftist tree is a priority queue. It is implemented with a variant of
the binary heap. We store the distance of the nearest leaf in the subtree rooted at a
node for every node. Let us call this value s-value. Leftist heap attempts to be very
unbalanced which is in contrast to a binary heap, which is always a complete binary
tree.
Also see, Implementation of Heap
A leftist heap is an implementation of a mergeable heap. While inserting a new node
into the heap, a new one-node heap is created, which is merged into the existing
heap. To delete an item, we replace it with the merge of its left and right subtrees.
Leftist heaps are useful because of their ability to merge quickly (in O(logN) time)
compared to binary heaps that take O(N) time.
Time Complexities
The following are the time complexities of various operations of a leftist heap:
Function Complexity
Get Min O(1)
Delete Min O(logN)
Insert O(logN)
Merge O(logN)
Properties
 The following are the properties of a leftist heap:
1. Key(i) >= Key(parent(i)): This is similar to normal min-heap property.
2. The right descendant of every node has a smaller s-value. This means that the
number of edges on the shortest path from a node to a leaf of the right child is
less than or equal to that of the left child.
 From the above properties, we can conclude the following:
1. The smallest path from any leaf to the root is the path from the root to the
rightmost leaf.
2. If there are ‘X’ nodes in the path from the root to the rightmost leaf, then the
leftist heap will have at least (2x
- 1) nodes.
3. From point 2, we can also conclude that the length of the path from the root to
the leftmost leaf is O(logN) for a leftist heap having N nodes.
Operations
1. merge(): The main operation of a leftist heap is the merge operation.
2. deleteMin(): This can be done by replacing the minimum node with the merge of
left and right subtrees.
3. insert(): This can be done by calling the merge function on the original heap and
a new one-node heap.
S-value or Dist
The s-value (or dist) of a node is the distance between that node and the nearest
leaf in the subtree of that node. The dist of a null child is 0 (In some implementations,
the dist of a null child is assumed to be -1).

2. Merging
Since the right subtree of a leftist heap is smaller than the left subtree, we merge the
right subtree with the other tree.
The following are the steps for merge operation:
1. Make the root with a smaller value than the new root.
2. Push this root into an empty stack and move to its right child.
3. Recursively compare the two keys, keep pushing the smaller key into the stack,
and move to its right child.
4. Perform step 3 until a null node is reached.
5. Pick the last node processed and make it the right child of the node present at
the top of the stack.
6. Recursively pop the elements from the stack and keep making them the right
child of the new node present at the stack top.
Merging visualization
Inserting into a leftist heap
The insertion operation is done using the merge operation. We create a new leftist
heap with a single node having a value of the element to be inserted. We then merge
this newly created heap with the existing heap.
1. newHeap = createLeftistHeap(val)
2. merge(leftistHeap, newHeap)
Deleting Min element from a leftist heap
The min-element is the root of a leftist heap. So, to delete the min-element, we
replace the root element with the merge of its left and right subtrees.
1. Val = root.val
2. root = merge(root.right, root.left)
3. Return val

3. Implementation
// C++ program for leftist heap / leftist tree
#include <bits/stdc++.h>
using namespace std;
// Defining class Node for leftist heap node.
class Node
{
public:
int val;
Node *left;
Node *right;
int dist;
Node(int &val, Node *lt = NULL, Node *rt = NULL, int np = 0)
{
this->val = val;
right = rt;
left = lt,
dist = np;
}
};
// Defining Leftist Heap functions.
class LeftistHeap
{
public:
LeftistHeap();
LeftistHeap(LeftistHeap &rhs);
~LeftistHeap();
bool isEmpty();
bool isFull();
int &findMin();
void Insert(int &x);
void deleteMin();
void deleteMin(int &minItem);
void makeEmpty();
void Merge(LeftistHeap &rhs);
LeftistHeap &operator=(LeftistHeap &rhs);
private:
Node *root;
Node *Merge(Node *h1,
Node *h2);
Node *Merge1(Node *h1,
Node *h2);
void swapChildren(Node *t);
void reclaimMemory(Node *t);
Node *clone(Node *t);
};

4. // Function to construct leftist heap
LeftistHeap::LeftistHeap()
{
root = NULL;
}
// Copy constructor.
LeftistHeap::LeftistHeap(LeftistHeap &rhs)
{
root = NULL;
*this = rhs;
}
// Destructor for the leftist heap
LeftistHeap::~LeftistHeap()
{
makeEmpty();
}
// Function to merge rhs into the priority queue.
void LeftistHeap::Merge(LeftistHeap &rhs)
{
if (this == &rhs)
return;
root = Merge(root, rhs.root);
rhs.root = NULL;
}
// Function to merge two roots
Node *LeftistHeap::Merge(Node *h1,
Node *h2)
{
if (h1 == NULL)
return h2;
if (h2 == NULL)
return h1;
if (h1->val < h2->val)
return Merge1(h1, h2);
else
return Merge1(h2, h1);
}
// Function to merge two roots.
Node *LeftistHeap::Merge1(Node *h1,
Node *h2)
{
if (h1->left == NULL)
h1->left = h2;
else
{

5. h1->right = Merge(h1->right, h2);
if (h1->left->dist < h1->right->dist)
swapChildren(h1);
h1->dist = h1->right->dist + 1;
}
return h1;
}
// Function to swap children of a node.
void LeftistHeap::swapChildren(Node *t)
{
Node *tmp = t->left;
t->left = t->right;
t->right = tmp;
}
// Function to insert an item into leftist heap
void LeftistHeap::Insert(int &x)
{
root = Merge(new Node(x), root);
}
// Function to find the smallest item.
int &LeftistHeap::findMin()
{
return root->val;
}
// Function to remove the smallest item.
void LeftistHeap::deleteMin()
{
Node *oldRoot = root;
root = Merge(root->left, root->right);
delete oldRoot;
}
// Function to remove the smallest item.
void LeftistHeap::deleteMin(int &minItem)
{
if (isEmpty())
{
cout << "Heap is Empty" << endl;
return;
}
minItem = findMin();
deleteMin();
}
// Function to test if the priority queue is logically empty.
bool LeftistHeap::isEmpty()

6. {
return root == NULL;
}
// Function to test if the priority queue is logically full.
bool LeftistHeap::isFull()
{
return false;
}
// Make the priority queue logically empty
void LeftistHeap::makeEmpty()
{
reclaimMemory(root);
root = NULL;
}
// Deep copy
LeftistHeap &LeftistHeap::operator=(LeftistHeap &rhs)
{
if (this != &rhs)
{
makeEmpty();
root = clone(rhs.root);
}
return *this;
}
// Internal method to make the tree empty.
void LeftistHeap::reclaimMemory(Node *t)
{
if (t != NULL){
reclaimMemory(t->left);
reclaimMemory(t->right);
delete t;
}
}
// Internal method to clone subtree.
Node *LeftistHeap::clone(Node *t)
{
if (t == NULL){
return NULL;
}
else{
return new Node(t->val, clone(t->left), clone(t->right), t->dist);
}
}
// Driver program

7. int main()
{
LeftistHeap h;
LeftistHeap h1;
LeftistHeap h2;
int x;
int arr[] = {1, 5, 7, 10, 15};
int arr1[] = {22, 75};
h.Insert(arr[0]);
h.Insert(arr[1]);
h.Insert(arr[2]);
h.Insert(arr[3]);
h.Insert(arr[4]);
h1.Insert(arr1[0]);
h1.Insert(arr1[1]);
h.deleteMin(x);
cout << x << endl;
h1.deleteMin(x);
cout << x << endl;
h.Merge(h1);
h2 = h;
h2.deleteMin(x);
cout << x << endl;
return 0;
}
Output
1
22
5