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# Perfect Binary Tree

#### In this tutorial, you will learn about the perfect binary tree. Also, you will find working examples for checking a perfect binary tree in C, C++, Java and Python.

A perfect binary tree is a type of binary tree in which every internal node has exactly two child nodes and all the leaf nodes are at the same level. Perfect Binary Tree

All the internal nodes have a degree of 2.

Recursively, a perfect binary tree can be defined as:

1. If a single node has no children, it is a perfect binary tree of height `h = 0`,
2. If a node has `h > 0`, it is a perfect binary tree if both of its subtrees are of height `h - 1` and are non-overlapping. Perfect Binary Tree (Recursive Representation)

## Python Examples

The following code is for checking whether a tree is a perfect binary tree.

``````/* Checking if a binary tree is a perfect binary tree in Python */

class newNode:
def __init__(self, k):
self.key = k
self.right = self.left = None

/* Calculate the depth */
def calculateDepth(node):
d = 0
while (node is not None):
d += 1
node = node.left
return d

/* Check if the tree is perfect binary tree */
def is_perfect(root, d, level=0):

/* Check if the tree is empty */
if (root is None):
return True

/* Check the presence of trees */
if (root.left is None and root.right is None):
return (d == level + 1)

if (root.left is None or root.right is None):
return False

return (is_perfect(root.left, d, level + 1) and
is_perfect(root.right, d, level + 1))

root = None
root = newNode(1)
root.left = newNode(2)
root.right = newNode(3)
root.left.left = newNode(4)
root.left.right = newNode(5)

if (is_perfect(root, calculateDepth(root))):
print("The tree is a perfect binary tree")
else:
print("The tree is not a perfect binary tree")``````

## Perfect Binary Tree Theorems

1. A perfect binary tree of height h has `2h + 1 – 1` node.
2. A perfect binary tree with n nodes has height `log(n + 1) – 1 = Θ(ln(n))`.
3. A perfect binary tree of height h has `2h` leaf nodes.
4. The average depth of a node in a perfect binary tree is `Θ(ln(n))`.

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