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CS3 Data Structures & Algorithms

Chapter 7 Binary Trees

Show Source |    | About   «  7.3. Binary Tree as a Recursive Data Structure   ::   Contents   ::   7.5. Binary Tree Traversals  »

7.4. The Full Binary Tree Theorem

Some binary tree implementations store data only at the leaf nodes, using the internal nodes to provide structure to the tree. By definition, a leaf node does not need to store pointers to its (empty) children. More generally, binary tree implementations might require some amount of space for internal nodes, and a different amount for leaf nodes. Thus, to compute the space required by such implementations, it is useful to know the minimum and maximum fraction of the nodes that are leaves in a tree containing \(n\) internal nodes.

Unfortunately, this fraction is not fixed. A binary tree of \(n\) internal nodes might have only one leaf. This occurs when the internal nodes are arranged in a chain ending in a single leaf as shown in Figure 7.4.1. In this example, the number of leaves is low because each internal node has only one non-empty child. To find an upper bound on the number of leaves for a tree of \(n\) internal nodes, first note that the upper bound will occur when each internal node has two non-empty children, that is, when the tree is full. However, this observation does not tell what shape of tree will yield the highest percentage of non-empty leaves. It turns out not to matter, because all full binary trees with \(n\) internal nodes have the same number of leaves. This fact allows us to compute the space requirements for a full binary tree implementation whose leaves require a different amount of space from its internal nodes.

Figure 7.4.1: A tree containing many internal nodes and a single leaf.

When analyzing the space requirements for a binary tree implementation, it is useful to know how many empty subtrees a tree contains. A simple extension of the Full Binary Tree Theorem tells us exactly how many empty subtrees there are in any binary tree, whether full or not. Here are two approaches to proving the following theorem, and each suggests a useful way of thinking about binary trees.

   «  7.3. Binary Tree as a Recursive Data Structure   ::   Contents   ::   7.5. Binary Tree Traversals  »

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