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Figure 11.7 Isomorphic and nonisomorphic trees
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Binary trees are ordered trees. The order of the two children at each node is part of the structure of the binary tree. Binary trees are ordered trees. So any isomorphism between binary trees must preserve the order of each node s children. EXAMPLE 11.9 Nonisomorphic Binary Trees
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Figure 11.8 Nonisomorphic binary trees
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In Figure 11.8, Binary Tree 1 is not isomorphic to Binary Tree 2, for the same reason that the ordered trees in Example 11.8 are not isomorphic: The subtrees don t all match, as ordered trees. In Tree 1, the root s right child has a left child; but in Tree 1, the root s right child has no (nonempty) left child.
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COMPLETE BINARY TREES A complete binary tree is either a full binary tree or one that is full except for a segment of missing leaves on the right side of the bottom level. EXAMPLE 11.10 A Complete Binary Tree of Height 3
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The tree shown in Figure 11.9 is complete. It is shown together with the full binary tree from which it was obtained by adding five leaves on the right at level 3.
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Figure 11.9 Complete binary trees
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Theorem 11.2 In a complete binary tree of height h, h + 1 n 2h+1 1 and h = lg n where n is the number of its nodes.
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[CHAP. 11
EXAMPLE 11.11 More Complete Binary Trees Figure 11.10 shows three more examples of complete binary trees.
Figure 11.10 Complete binary trees
Complete binary trees are important because they have a simple and natural implementation using ordinary arrays. The natural mapping is actually defined for any binary tree: Assign the number 1 to the root; for any node, if i is its number, then assign 2i to its left child and 2i+1 to its right child (if they exist). This assigns a unique positive integer to each node. Then simply store the element at node i in a[i], where a[] is an array. Complete binary trees are important because of the simple way in which they can be stored in an array. This is achieved by assigning index numbers to the tree nodes by level, as shown in Figure 11.11. The beauty in this natural mapping is the simple way that it allows the array indexes of the children and parent of a node to be computed.
Figure 11.11 The natural mapping of a complete binary tree
Algorithm 11.1 The Natural Mapping of a Complete Binary Tree into an Array To navigate about a complete binary tree stored by its natural mapping in an array: 1. The parent of the node stored at location i is stored at location i/2. 2. The left child of the node stored at location i is stored at location 2i. 3. The right child of the node stored at location i is stored at location 2i + 1. For example, node e is stored at index i = 5 in the array; its parent node b is stored at index i/2 = 5/2 = 2, its left child node j is stored at location 2i = 2 5 = 10, and its right child node k is stored at index 2i + 1 = 2 5 + 1 = 11. The use of the adjective complete should now be clear: The defining property for complete binary trees is precisely the condition that guarantees that the natural mapping will store the tree nodes completely in an array with no gaps.
CHAP. 11]
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EXAMPLE 11.12 An Incomplete Binary Tree
Figure 11.12 shows the incomplete binary tree from Example 11.1 on page 201. The natural mapping of its nodes into an array leaves some gaps, as shown in Figure 11.13.
Note: Some authors use the term almost complete binary tree for a complete binary tree and the term complete binary tree for a full binary tree.
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