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Figure 11.40 shows how the forest that produced the specified binary tree was obtained by reversing the natural map.
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Figure 11.40 Mapping a forest into a binary tree
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f(h) = h + 1 f(h) = 1 a.
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public int leaves() { if (this == null) { return 0; } int leftLeaves = (left==null 0 : left.leaves()); int rightLeaves = (right==null 0 : right.leaves()); return leftLeaves + rightLeaves; } public int height() { if (this == null) { return -1; } int leftHeight = (left==null -1 : left.height()); int rightHeight = (right==null -1 : right.height()); return 1 + (leftHeight<rightHeight rightHeight : leftHeight); } public int level(Object object) { if (this == null) { return -1; } else if (object == root) { return 0; } int leftLevel = (left==null -1 : left.level(object)); int rightLevel = (right==null -1 : right.level(object)); if (leftLevel < 0 && rightLevel < 0) { return -1; } return 1 + (leftLevel<rightLevel rightLevel : leftLevel); } public void reflect() { if (this == null) { return; } if (left != null) { left.reflect(); } if (right != null) { right.reflect(); } BinaryTree temp=left; left = right; right = temp; }
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public void defoliate() { if (this == null) { return; } else if (left == null && right == null) { root = null; return; } if (left != null && left.left==null && left.right==null) { left = null; } else { left.defoliate(); } if (right != null && right.left==null && right.right==null) right = null; } else { right.defoliate(); } }
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Search Trees
Tree structures are used to store data because their organization renders more efficient access to the data. A search tree is a tree that maintains its data in some sorted order. MULTIWAY SEARCH TREES Here is the recursive definition of a multiway search tree: A multiway search tree of order m is either the empty set or a pair (k, S), where the first component is a sequence k = (k1, k2, . . ., kn 1) of n 1 keys and the second component is a sequence S = (S0, S1, S2, . . ., Sn 1) of n multiway search trees of order m, with 2 n m, and s0 k1 s1 . . . kn 1 sn 1 for each si Si. This is similar to the recursive definition of a general tree on page 186. A multiway search tree of order m can be regarded as a tree of order m in which the elements are sequences of keys with the ordering property described above. EXAMPLE 12.1 A Five-Way Search Tree
Here is an m-way search tree with m = 5. It has three internal nodes of degree 5 (each containing four keys), three internal nodes of degree 4 (each containing three keys), four internal nodes of degree 3 (each containing two keys), and one internal node of degree 2 (containing one key).
Figure 12.1 A five-way search tree
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The root node has two keys and three children. All four keys in the first child are less than k 1 = 57. All three keys in the second child are between k 1 = 57 and k 2 = 72. Both keys in the third child are greater than k 2 = 72. In fact, all thirteen keys in the first subtree are less than 57, all seven keys in the second subtree are between 57 and 72, and all eight keys in the third subtree are greater than 72.
An m-way search tree is called a search tree because it serves as a multilevel index for searching large lists. To search for a key value, begin at the root and proceed down the tree until the key is found or a leaf is reached. At each node, perform a binary search for the key. It it is not found in that node, the search will stop between two adjacent key values (with k 0 = and k n = ). In that case, follow the link that is between those two keys to the next node. If we reach a leaf, then we know that the key is not in the tree. For example, to search for key value 66, start at the root of the tree and then follow the middle link (because 57 66 < 72) down to the middle three-key node. Then follow its third link (because 60 66 < 70) down to the bottom four-key node. Then follow its third link (because 65 66 < 67) down to that leaf node. Then conclude that the key 66 is not in the tree. To insert a key into an m-way search tree, first apply the search algorithm. If the search ends at a leaf node, then the two bracketing keys of its parent node locate the correct position for the new key. So insert it in that internal node between those two bracketing keys. If that insertion gives the node m keys (thereby exceeding the limit of m 1 keys per node), then split the node into two nodes after moving its middle key up to its parent node. If that move gives the parent node m keys, repeat the splitting process. This process can iterate all the way back up to the root, if necessary. Splitting the root produces a new root, thereby increasing the height of the tree by one level. EXAMPLE 12.2 Inserting into a Five-Way Tree
To insert 66 into the search tree of Example 12.1, first perform the search, as described above. This leads to the leaf node marked with an X in Figure 12.2:
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