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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.

All the internal nodes have a degree of 2.

Recursively, a perfect binary tree can be defined as:

- If a single node has no children, it is a perfect binary tree of height
`h = 0`

, - 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.

## 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

- A perfect binary tree of height h has
`2`

node.^{h + 1}– 1 - A perfect binary tree with n nodes has height
`log(n + 1) – 1 = Θ(ln(n))`

. - A perfect binary tree of height h has
`2`

leaf nodes.^{h} - The average depth of a node in a perfect binary tree is
`Θ(ln(n))`

.

# Python Example for Beginners

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