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Thus, by the Principle of Mathematical Induction (see page 321), the formula must be correct for all full trees of any height. 10.3 Proof of Corollary 10.1 on page 188: This proof is purely algebraic: dh + 1 1 n = -------------------d 1 n d 1 = dh + 1 1 dh + 1 = n d 1 + 1 = nd n + 1 h + 1 = log d nd n + 1 h
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Proof of Corollary 10.2 on page 188: Let T be a tree of any order d and any height h. Then T can be embedded into the full tree of the h+1 1 same degree and height. That full tree has exactly d -------------------- nodes, so its subtree T has at most that d 1 many nodes.
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The path length of a full tree of order d and height h is ------------------ h d h + 1 h + 1 d + 1 . For example, 2 d 102. the path length of the full tree on Figure 10.4 on page 188 is 1 The tree diagram analysis of the St. Petersburg Paradox is shown in Figure 10.15. The flaw in this strategy is that there is a distinct possibility (i.e., a positive probability) that enough tails could come up in a row to make the required bet exceed the bettor s stake. After n successive tails, the bettor must bet \$2n. For example, if 20 tails come up in a row, the next bet will have to be more than a million dollars!
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Figure 10.15 Analysis of the St. Petersburg Paradox
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The decision tree for the version of craps where 3 can be a point is shown in Figure 10.16. The probability that X wins this version is 0.5068 or 50.68 percent. The decision tree in Figure 10.17 shows all possible outcomes from the algorithm that solves the 7coin problem.
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Figure 10.16 Decision tree for a craps game
Figure 10.17 Decision tree for the 7-coin problem
Binary Trees
DEFINITIONS Here is the recursive definition of a binary tree: A binary tree is either the empty set or a triple T = (x, L, R), where x is a node and L and R are disjoint binary trees, neither of which contains x. The node x is called the root of the tree T, and the subtrees L and R are called the left subtree and the right subtree of T rooted at x. Comparing this definition with the one on page 186, it is easy to see that a binary tree is just an ordered tree of order 2. But be aware that an empty left subtree is different from an empty right subtree. (See Example 10.5 on Figure 11.1 Unequal binary trees page 191.) Consequently, the two binary trees shown Figure 11.1 are not the same. Here is an equivalent, nonrecursive definition for binary trees: A binary tree is an ordered tree in which every internal node has degree 2. In this simpler definition, the leaf nodes are regarded as dummy nodes whose only purpose is to define the structure of the tree. In applications, the internal nodes would hold data, while the leaf nodes would be either identical empty nodes, a single empty node, or just the null reference. This may seem inefficient and Figure 11.2 Equal binary trees more complex, but it is usually easier to implement. In Figure 11.2, the dummy leaf nodes in the tree on the right are shown as asterisks.
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Except where noted, in this book we adhere to the first definition for binary trees. So some internal nodes may have only one child, either a left child or a right child. The definitions of the terms size, path, length of a path, depth of a node, level, height, interior node, ancestor, descendant, subtree, and supertree are the same for binary trees as for general trees. (See page 186.) EXAMPLE 11.1 Characteristics of a Binary Tree
Figure 11.3 shows a binary tree of size 10 and height 3. Node a is its root. The path from node h to node b has length 2. Node b is at level 1, and node h is at level 3. b is an ancestor of h, and h is a descendant of b. The part in the shaded region is a subtree of size 6 and height 2. Its root is node b.