Data Structures and Algorithms with Object-Oriented Design Patterns in Java

Binary Trees

In this section we consider an extremely important and useful category of tree structure--binary trees . A binary tree is an N-ary tree for which N is two. Since a binary tree is an N-ary tree, all of the results derived in the preceding section apply to binary trees. However, binary trees have some interesting characteristics that arise from the restriction that N is two. For example, there is an interesting relationship between binary trees and the binary number system. Binary trees are also very useful for the representation of mathematical expressions involving the binary operations such as addition and multiplication.

Binary trees are defined as follows:

Definition (Binary Tree) A binary tree T is a finite set of nodes with the following properties:
Either the set is empty, ; or
The set consists of a root, r, and exactly two distinct binary trees and , .
The tree is called the left subtree of T, and the tree is called the right subtree of T.

Binary trees are almost always considered to be ordered trees . Therefore, the two subtrees and are called the left and right subtrees, respectively. Consider the two binary trees shown in Figure . Both trees have a root with a single non-empty subtree. However, in one case it is the left subtree which is non-empty; in the other case it is the right subtree that is non-empty. Since the order of the subtrees matters, the two binary trees shown in Figure are different.

Figure: Two distinct binary trees.

We can determine some of the characteristics of binary trees from the theorems given in the preceding section by letting N=2. For example, Theorem tells us that an binary tree with internal nodes contains n+1 external nodes. This result is true regardless of the shape of the tree. Consequently, we expect that the storage overhead of associated with the empty trees will be O(n).

From Theorem we learn that a binary tree of height has at most internal nodes. Conversely, the height of a binary tree with n internal nodes is at least . That is, the height of a binary tree with n nodes is .

Finally, according to Theorem , a binary tree of height has at most leaves. Conversely, the height of a binary tree with l leaves is at least . Thus, the height of a binary tree with l leaves is