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Complete binary tree and heap
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- 3. 3 Complete Binary Tree A complete binary tree is a tree that is completely filled, with the possible exception of the bottom level. The bottom level is filled from left to right. Different definition of complete binary tree than an earlier definition we had. The earlier definition could be called a perfect binary tree.
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- 5. 5 Complete Binary Tree Recall that such a tree of height h has between 2h to 2h+1 –1 nodes. Because the tree is so regular, it can be stored in an array; no pointers are necessary.
- 6. 6 Complete Binary Tree A CB E I D HJ GF A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 7. 7 Complete Binary Tree For any array element at position i, the left child is at 2i, the right child is at (2i +1) and the parent is at floor(i /2). A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 8. 8 Complete Binary Tree For any array element at position i, the left child is at 2i, the right child is at (2i +1) and the parent is at i 2. A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 9. 9 Complete Binary Tree For any array element at position i, the left child is at 2i, the right child is at (2i +1) and the parent is at i 2. A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 10. 10 Complete Binary Tree For any array element at position i, the left child is at 2i, the right child is at (2i +1) and the parent is at i 2. A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 11. 11 Complete Binary Tree For any array element at position i, the left child is at 2i, the right child is at (2i +1) and the parent is at i 2. A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 12. 12 Complete Binary Tree Level-ordernumbers array index A CB E I D H J GF A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 4 5 6 7 8 9 10
- 13. 13 Complete Binary Tree A CB E I D HJ GF 1 2 3 4 5 6 7 8 9 10 A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140
- 14. 14 Complete Binary Tree Question: why don’t we store all binary trees in arrays? Why use pointers? A CB E I D H J GF A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 4 5 6 7 8 9 10
- 15. 15 Complete Binary Tree Question: why don’t we store all binary trees in arrays? Why use pointers? A CB E I D H J GF A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 4 5 6 7 8 9 10
- 16. 16 Complete Binary Tree Question: why don’t we store all binary trees in arrays? Why use pointers? A CB E I D H J GF A B C D E F G H I J 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 4 5 6 7 8 9 10
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- 18. 18 Heap A heapis a complete binary tree that conforms to the heap order. The heap order property: in a (min) heap, for every node X, the key in the parent is smaller than (or equal to) the key in X. Or, the parent node has key smaller than or equal to both of its children nodes.
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- 20. 20 Heap Not a heap:heap property violated 13 1921 31 26 6 65 32 6816
- 21. 21 Heap Analogously, wecan define a max-heap, where the parent has a key larger than the its two children. Thus the largest key would be in the root.
- 22. 22 Inserting into aHeap 13 1621 31 26 24 65 32 6819 13 21 16 24 31 19 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 This is an existing heap
- 23. 23 Inserting into aHeap insert(14) 13 1621 31 26 24 65 32 6819 13 21 16 24 31 19 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 1411
- 24. 24 Inserting into aHeap insert(14) 13 1621 31 26 24 65 32 6819 13 21 16 24 31 19 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 11
- 25. 25 Inserting into aHeap insert(14) 13 1621 3126 24 65 32 6819 13 21 16 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 11
- 26. 26 Inserting into aHeap insert(14) 13 16 21 3126 24 65 32 6819 13 2116 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 11
- 27. 27 Inserting into aHeap insert(14) 13 16 21 3126 24 65 32 6819 13 2116 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 11 14 14
- 28. 28 Inserting into aHeap insert(14) with exchange 13 1621 31 26 24 65 32 6819 13 21 16 24 31 19 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 1411
- 29. 29 Inserting into aHeap insert(14) with exchange 13 1621 3126 24 65 32 6819 13 21 16 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11
- 30. 30 Inserting into aHeap insert(14) with exchange 13 16 21 3126 24 65 32 6819 13 2116 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11
- 31. 31 Inserting into aHeap insert(15) with exchange 13 16 21 3126 24 65 32 6819 13 2116 24 3119 68 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11 15 15 12
- 32. 32 Inserting into aHeap insert(15) with exchange 13 16 21 3126 24 65 32 68 19 13 2116 24 31 1968 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11 15 15 12
- 33. 33 Inserting into aHeap insert(15) with exchange 13 1621 3126 24 65 32 68 19 13 21 1624 31 1968 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11 15 15 12
- 34. 34 Inserting into aHeap insert(15) with exchange 13 1621 3126 24 65 32 68 19 13 21 1624 31 1968 65 26 32 1 2 3 4 5 6 7 8 9 10 11 12 13 140 1 2 3 7654 8 9 10 14 14 11 15 15 12
- 35. 35 DeleteMin Finding theminimum is easy; it is at the top of the heap. Deleting it (or removing it) causes a hole which needs to be filled. 13 16 21 3126 24 65 32 6819 1 2 3 7654 8 9 10 14 11
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- 40. 40 DeleteMin deleteMin(): heap sizeis reduced by 1. 14 16 31 26 24 65 32 6819 1 2 3 7654 8 9 10 21