Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 5.1 Definitions and Properties 147AABB(a)(b)AABEMPTYEMPTYB(c)(d)Figure 5.2 Two different binary trees. (a) A binary tree whose root has a nonemptyleft child. (b) A binary tree whose root has a non-empty right child. (c) Thebinary tree of (a) with the missing right child made explicit. (d) The binary treeof (b) with the missing left child made explicit.(a)(b)Figure 5.3 Examples of full and complete binary trees. (a) This tree is full (butnot complete). (b) This tree is complete (but not full).not full. The heap data structure (Section 5.5) is an example of a complete binarytree. The Huffman coding tree (Section 5.6) is an example of a full binary tree.5.1.1 The Full Binary Tree TheoremSome binary tree implementations store data only at the leaf nodes, using the internalnodes to provide structure to the tree. More generally, binary tree implementationsmight require some amount of space for internal nodes, and a different amountfor leaf nodes. Thus, to analyze the space required by such implementations, it isuseful to know the minimum and maximum fraction of the nodes that are leaves ina tree containing n internal nodes.Unfortunately, this fraction is not fixed. A binary tree of n internal nodes mighthave only one leaf. This occurs when the internal nodes are arranged in a chainending in a single leaf as shown in Figure 5.4. In this case, the number of leavesis low because each internal node has only one non-empty child. To find an upperbound on the number of leaves for a tree of n internal nodes, first note that the upper