We have discussed Introduction to Binary Tree in set 1. In this post, the properties of a binary tree are discussed.
1) The maximum number of nodes at level ‘l’ of a binary tree is 2l.
Here level is the number of nodes on the path from the root to the node (including root and node). Level of the root is 0.
This can be proved by induction.
For root, l = 0, number of nodes = 20 = 1
Assume that the maximum number of nodes on level ‘l’ is 2l
Since in Binary tree every node has at most 2 children, next level would have twice nodes, i.e. 2 * 2l
2) The Maximum number of nodes in a binary tree of height ‘h’ is 2h – 1.
Here the height of a tree is the maximum number of nodes on the root to leaf path. Height of a tree with a single node is considered as 1.
This result can be derived from point 2 above. A tree has maximum nodes if all levels have maximum nodes. So maximum number of nodes in a binary tree of height h is 1 + 2 + 4 + .. + 2h-1. This is a simple geometric series with h terms and sum of this series is 2h – 1.
In some books, the height of the root is considered as 0. In this convention, the above formula becomes 2h+1 – 1
3) In a Binary Tree with N nodes, minimum possible height or the minimum number of levels is? Log2(N+1) ?
This can be directly derived from point 2 above. If we consider the convention where the height of a leaf node is considered as 0, then above formula for minimum possible height becomes? Log2(N+1) ? – 1
4) A Binary Tree with L leaves has at least? Log2L? + 1 levels
A Binary tree has the maximum number of leaves (and a minimum number of levels) when all levels are fully filled. Let all leaves be at level l, then below is true for the number of leaves L.
L <= 2l-1 [From Point 1] l = ? Log2L ? + 1 where l is the minimum number of levels.
5) In Binary tree where every node has 0 or 2 children, the number of leaf nodes is always one more than nodes with two children.
L = T + 1 Where L = Number of leaf nodes T = Number of internal nodes with two children
See Handshaking Lemma and Tree for proof.
In the next article on tree series, we will be discussing different types of Binary Trees and their properties.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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- Complexity of different operations in Binary tree, Binary Search Tree and AVL tree
- Handshaking Lemma and Interesting Tree Properties
- Check if a binary tree is subtree of another binary tree | Set 1
- Check if a binary tree is subtree of another binary tree | Set 2
- Convert a Binary Tree to Threaded binary tree | Set 1 (Using Queue)
- Convert a Binary Tree to Threaded binary tree | Set 2 (Efficient)
- Binary Tree to Binary Search Tree Conversion using STL set
- Binary Tree | Set 3 (Types of Binary Tree)
- Maximum sub-tree sum in a Binary Tree such that the sub-tree is also a BST
- Convert a Generic Tree(N-array Tree) to Binary Tree
- Binary Tree to Binary Search Tree Conversion
- Check whether a binary tree is a full binary tree or not
- Minimum swap required to convert binary tree to binary search tree
- Check whether a binary tree is a full binary tree or not | Iterative Approach
- Check whether a given binary tree is skewed binary tree or not?
- Difference between Binary Tree and Binary Search Tree
- Check if a binary tree is subtree of another binary tree using preorder traversal : Iterative
- Check whether a binary tree is a complete tree or not | Set 2 (Recursive Solution)
- Print Binary Tree levels in sorted order | Set 3 (Tree given as array)
- Check if the given binary tree has a sub-tree with equal no of 1's and 0's | Set 2