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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.

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x I( k ) =

k = Iandk=N-1

private static void merge(int[] a, int p, int m, int q) { // PRECONDITIONS: 0 <= p <= m < q <= a.length; 20 // a[p..m) is sorted; 21 // a[m..q) is sorted; 22 // POSTCONDITION: a[p..q) is sorted; 23 if (a[m-1] <= a[m]) { 24 return; 25 } 26 int i = p, j = m, k = 0; 27 int[] tmp = new int[q-p]; 28 while (i < m && j < q) { 29 // INVARIANT: tmp[0..k) is sorted 30 tmp[k++] = ( a[i]<=a[j] a[i++] : a[j++] ); 31 } 32 System.arraycopy(a, i, a, p+k, m-i); 33 System.arraycopy(tmp, 0, a, p, k); 34 } The main sort() method sorts the entire array by invoking the overloaded sort() method with

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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.

Therefore,

18 19

and the inverse DFT is a sinusoid with an amplitude of N / 2 :

CHAP. 11]

x(n) =xl(n)

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parameters for the starting index k and the length n of the subarray. That three-parameter method sorts the subarray by sorting its left half and its right half separately and then merging them. The merge() method merges the two halves a[p..m) and a[m..q) into a temporary array, where m is the middle index m = p + n/2. The while loop copies one element on each iteration; it copies the smaller of the two elements a[i] and a[j]. The post increment operator automatically advances the index of the copied element. When all the elements of one half have been copied, the while loop stops and then all the elements are copied back into a[].

o x2(n) =

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.

(2Y)

Theorem 149 The merge sort runs in O(n lgn) time In general, the merge sort works by repeatedly dividing the array in half until the pieces are singletons, and then it merges the pieces pairwise until a single piece remains This is illustrated by the diagram in Figure 141 The number of iterations in the first part equals the number of times n can be halved: that is, lg n 1 In terms of the number and sizes of the pieces, the second part of the process reverses the first So the second part also has lg n 1 steps So the entire algorithm has O(lg n) steps Each step compares all n elements So the total number of comparisons is O(n lg n) Theorem 1410 The merge sort is correct The proof follows from the preconditions and postconditions given in the code In the Figure 14.

Therefore,

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.

r . r , x 2 ( n= - i N sin )

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