### Table 3: Binary search tree

"... In PAGE 17: ... Table3 shows that if the elements are inserted in ascending order, the constructed tree is in fact a linear list where the element representing the minimum is at the root and the maximum at the end. The method getMax() is called twice in the postcondition, once to get the maximum of the old tree and once to get the maximum of the tree after inserting the new element.... ..."

### Table III. Type inference of Binary Search Trees

### Table 2: Examples of notations for a binary search tree in Figure 1.1

2006

### Table 2.4: Number binary symbols coded using trees created from difierent types of binary searches, applied to data source of Example 10. The trees corresponding to bisection and optimal searches are shown in Figures 2.3 and 2.4, respectively.

2004

### Table 1: Main results obtained with fringe analysis. Some attempts to include deletions into the analysis had been made for AVL-trees [Meh82], and B-trees [Miz79, QK80, BY87a, JS89]. However, the random model does not apply after deletions, and a di erent model must be used. In this review we give an unifying view to several variations of fringe analysis, and we present the complete solution for a fringe analysis type recurrence. Some theorems and proofs are included to keep the text self-contained and/or because they are contained in papers that are di cult to obtain. Fringe type recurrences can be applied to the analysis of any class of search trees, ranging from binary search trees to B-trees. The asymptotic solution shows a fast convergence of the process 3

"... In PAGE 3: ... Also it is possible to obtain bounds for the height of the tree [Ziv82, EZG+82, BY85] or probabilities related to concurrent access to the tree [KW80, EZG+82, BYP85]. Table1 shows some upper and lower bounds for the fraction of height balanced nodes and the probability of a rotation in AVL trees, and the fraction of internal nodes (with respect to all the n keys stored) and the probability of a split in 2-3 trees and B-trees of order m (internal nodes can hold between m and 2m keys). We also include the storage utilization for the case of 2-3 and B-trees.... ..."

### Table 2. Subject structures and their integrity constraints. Subjects vary in the complexity of their structural and data constraints. We generated a valid set of structures for each subject, and randomly injected faults that violate the integrity constraints.

"... In PAGE 9: ... 5.2 Subjects Table2 lists the subject structures and the integrity con- straints. The binary search tree and the doubly linked list are both presented in Section 2.... ..."

### Table 17: Run time on random inputs using integer keys (version 1 code) given in Tables 19 and 20. The sum of the run time for parts (b) and (d) of the experiment is graphed in Figure 15 for random data and in Figure 16 for ordered data. The graph of Figure 17 shows only one line MIX for AVL, RB-T, RB-B, WB, and BBST while that of Figure 18 shows MIX for AVL, RB-T, RB-B, and WB as the times for these are very close. With integer keys and random data, unbalanced binary search trees (BSTs) outperformed each of the remaining structures. The next best performance was exhibited by bottom-up red-black trees. They did marginally better than AVL trees. The remaining structures have a noticeably inferior structure. For ordered integer keys, BSTs take more time than we were willing to expend. Of the remaining structures, treaps generally performed best on parts (a), (c), and (e) while BBSTs did best on parts (b) and (d). 50

"... In PAGE 47: ... In fact, BBSTs perform about twice as many rotations as AVL trees. The average run times for the random data tests are given in Table17 and in Table 18 for the ordered data test. Both of these use integer keys.... ..."

Cited by 1

### Table 17: Run time on random inputs using integer keys (version 1 code) given in Tables 19 and 20. The sum of the run time for parts (b) and (d) of the experiment is graphed in Figure 15 for random data and in Figure 16 for ordered data. The graph of Figure 17 shows only one line MIX for AVL, RB-T, RB-B, WB, and BBST while that of Figure 18 shows MIX for AVL, RB-T, RB-B, and WB as the times for these are very close. With integer keys and random data, unbalanced binary search trees (BSTs) outperformed each of the remaining structures. The next best performance was exhibited by bottom-up red-black trees. They did marginally better than AVL trees. The remaining structures have a noticeably inferior structure. For ordered integer keys, BSTs take more time than we were willing to expend. Of the remaining structures, treaps generally performed best on parts (a), (c), and (e) while BBSTs did best on parts (b) and (d). 50

"... In PAGE 47: ... In fact, BBSTs perform about twice as many rotations as AVL trees. The average run times for the random data tests are given in Table17 and in Table 18 for the ordered data test. Both of these use integer keys.... ..."

Cited by 1

### Table 3 shows that if the elements are inserted in ascending order, the constructed tree is in fact a linear list where the element representing the minimum is at the root and the maximum at the end. The method getMax() is called twice in the postcondition, once to get the maximum of the old tree and once to get the maximum of the tree after inserting the new element. This explains the huge overhead. The evaluation of a balanced binary tree is the content of the next test (see table 4). The invariant ensures that the tree always is balanced.

"... In PAGE 16: ... insert original constrained code empty constr. strategy [ms] disabled [ms] enabled [ms] enabled [ms] perfect 26 26 (+ 0%) 200 (+ 669%) 93 (+ 257%) random 30 30 (+ 0%) 250 (+ 733%) 120 (+ 300%) ascending 90 110 (+22%) 125020 (+138811%) 2249 (+2398%) Table3 : Binary search tree 72 JOURNAL OF OBJECT TECHNOLOGY... ..."