Lecture4b dynamic data_structure
- 1. Lecture 4 Dynamic Data Structure Part 2 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 1
- 2. Content Lecture 4 Dynamic Data Part 2 Tree structures General Tree Representation Balanced Trees Binary Tree Sorted Binary Tree Binary Search Tree Insert an element in a Sorted Binary Tree Delete a node in a Sorted Binary Tree Other Tree Types 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 2
- 3. Tree structures General Trees are one of the most important data structures There are several different variants We will have a closer look to some of them Trees are used in computer science for data storage, searching and sorting 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 3
- 4. Tree structures Definition A Tree is a finite amount T of one or several nodes such that There is a significant node called root(T) The other nodes can be divided in disjunctive amount T1, .. Tm Each of these amounts is again a Tree These Trees are called Sub Trees This is a recursive definition At the end a Tree with only one node remains 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 4
- 5. Tree Representation You can display Trees in different ways In this course outline Trees are illustrated with the root on the top and the Sub Trees below This is a common illustration 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 5 FED B C A
- 6. Tree Representation These are some different representations 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 6
- 7. Tree Representation Family Trees A common example for trees are Family Trees Two variants of Family Trees exist: Starting from an ancestor as root and illustrated all the descendants Starting from a descendant as root and illustrated all the ancestors In Family Trees you can have redundancies if some of the ancestors have the same ancestors In this case the entries represent the role of the ancestor (for example grandmother on the mother’s side) 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 7
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- 10. Tree structures Definitions for Trees Each node in a Tree is the root of one of the Sub Trees Each Tree has zero or more child nodes which are below in the Tree A node with a child node is called parent node to the child (also ancestor node or superior) A root node is a node with no parents Each Tree has at least one root node 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 10
- 11. Tree structures Example 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 11 G FED B C A root node parent node of D, E, Fchild node of A
- 12. Tree structures The depth of a node is the length of the path to its root A node p is an ancestor to node q if p exists on the path from q to root The node q is called a descendant of p 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 12
- 13. Tree structures Example 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 13 G FED B C A depth of F is 2 C is ancestor of D D is a descendant of C
- 14. Tree structures The size of a node is the number of descendants a node has including itself Siblings are nodes that share the same parent node Nodes without a child node are called leaf node or terminal nodes Internal or inner node are nodes with one or more child nodes therefore size > 0 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 14
- 15. Tree structures Example 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 15 G FED B C A size of A is 7 B and C are siblings G is a leave node F is an inner node
- 16. Tree structures In-degree of a node is the number of edges arriving at that node The only node with In-degree = 0 is the root node Out-degree of a node is the number of edges leaving that node 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 16
- 17. Tree structures Example 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 17 G FED B C A In-degree of C 1 Out-degree of C is 3 In-degree of D 1 Out-degree of D is 0
- 18. Tree structures The level of a node is defined as: The level of the root(T) is 0 The level of any other node is increased by 1 to the level of the node which is the root of the superior sub tree level(node) = level(parent) + 1 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 18 G FED B C A Level 0 Level 1 Level 2 Level 3
- 19. Balanced Trees If the relative order of the Sub Trees is important you call it a Balanced Tree If the order is not important this is called an Oriented Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 19 G FED B C A G EFD B C A These Trees are different if you look at them as Balanced Tree but they are consider as the same if you look at them as Oriented Trees
- 20. Binary Trees A Binary Tree is a finite amount of nodes where the amount is empty or contains a root and two disjunctive Binary Trees These two Binary Trees are called left and right Sub Tree of the root That means that every node of the tree has zero, one or two child nodes Binary Trees are not a special case of Trees in general For example you can have an empty amount as a Binary Tree but not as a General Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 20
- 21. Binary Trees Any data in the Tree structure can be reached by starting at the rood node and following either the left or the right child Binary Trees are used for implementation of Binary Tree Search and Binary Heaps Examples for Binary Trees are the elimination contest in tennis or other sport competitions 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 21
- 22. Binary Trees Example These two Binary Trees are not the same One has left and the other a right Sub Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 22 B A B A Binary tree with a left sub tree Binary tree with a right sub tree
- 23. Binary Trees Implementation Binary Trees are easily implemented as a data type like One pointer points to the root If this pointer is null than the Tree is empty A node in the Binary Tree contains a pointer to an object or a record and the two pointers to the left and the right Sub Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 23
- 24. Binary Trees 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 24 root A B C E FD G H J ǁ ǁ ǁ ǁ ǁ ǁ ǁ ǁ ǁ ǁ
- 25. Binary Trees There are three possibilities to traverse a Binary Tree: preorder, inorder and postorder If the Tree is empty there is nothing to do Otherwise the traverse possibilities are defined as: 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 25 preorder visit the root traverse the left sub tree traverse the right sub tree inorder traverse the left sub tree visit the root traverse the right sub tree postorder traverse the left sub tree traverse the right sub tree visit the root B C A A B C B A C B C A
- 26. Binary Trees Examples 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 26 preorder: A B D C E G F H I inorder: D B A E G C H F I postorder: D B G E H I F C A G FED B C A IH Note The preorder traverse is also called Polish notation (after the Polish logician Jan Lukasiewicz) and the postorder traverse is therefore called a backwards Polish notation
- 27. Binary Trees Mathematical formulas can be represented by Binary Trees: 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 27 preorder - * * + 4 2 7 + 9 3 / 8 – 4 inorder (4 + 2) * 7 * (9 + 3) – (8/(-4)) (adding parentheses) postorder 4 2 + 7 * 9 3 + * 8 4 - / - -8* * / - 4+ 7 4 2 + 9 3
- 28. Sorted Binary Tree A tree T is a Sorted Binary Tree where v: T V function to receive the value of a node T a Binary Tree T1 the left Sub Tree (also a Sorted Binary Trees) T2 the right Sub Tree (also a Sorted Binary Trees) such that: v(root(T)) > v(t) for all nodes t in T1 v(root(T)) ≤ v(t) for all nodes t in T2 If you traverse a Sorted Binary Tree inorder, than all values are reached in a sorted way 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 28
- 29. Sorted Binary Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 29 7 961 3 8 5 Example Inorder: 1 3 5 6 7 8 9
- 30. Binary Tree Definitions The height of a Tree is the length of the path from the root to the deepest node in the Tree: h(T) = 0 if the Tree T is empty h(T) = max(h(T1), h(T2)) + 1 where T1 is the left and T2 the right Sub Tree The cardinality of a Tree T is the number n of elements in the Tree that means: n = card(T) 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 30
- 31. Binary Tree Example The height of this tree is 4 The cardinality of this tree is 7 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 31 7 961 3 8 5
- 32. Binary Tree A Tree is called height-balancing if the Tree T is empty or if | h(T1) – h (T2) | ≤ 1 where h is the height of a Tree and also the Sub Trees T1 and T2 are height-balancing Trees A Tree is called balanced if the Tree T is empty or if | card(T1) – card(T2) | ≤ 1 and also the Sub Trees T1 and T2 are balanced If a Tree T is balance he is also balanced in its height 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 32
- 33. Sorted Binary Tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 33 height-balancing Tree but not a balanced height left tree = 2, height right tree = 3 |h(T1) – h(T2) = |2 – 3| = 1 card left tree = 2, card right tree = 4 |card(T1) – card(T2) = |2 – 4| = 2 Balanced Tree ( height-balancing) height left tree = 2, height right tree = 2 |h(T1) – h(T2) = |2 – 2| = 0 card left tree = 3, card right tree = 3 |card(T1) – card(T2) = |3 – 3| = 0 7 961 3 8 5 751 2 6 4 3
- 34. Binary Search Tree k a value and the s search element v(s) = k T (Binary Tree) with t = root(T) If T is empty than there is nothing to do If v(t) = k than the search element is found If not than Replace T with the left subtree T1 if k < v(t) T = T1 Replace T with the right subtree T2 if k ≥ v(t) T = T2 The effort for the search is O(h(T)) 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 34
- 35. Binary Search Tree Example Search element k k = 1 t = 5 1 < 5 t = 3 1 < 3 t = 1 1 = 1 element found! k = 7 t = 5 7 > 5 t = 8 7 < 8 t = 6 7 > 6 t = 7 7 = 7 element found! 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 35 7 961 3 8 5
- 36. Insert a element in a Sorted Binary Tree u a values with k = v(u) to be insert T a tree with t the root of Tree T Create a new binary sorted tree U containing only the root u If T is empty replace T by U Otherwise replace T with the left sub tree T1 if k < v(t) T = T1 replace T with the right sub tree T2 if k ≥ v(t) T = T2 The effort is also O(h(T)) 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 36
- 37. Insert a element in a Sorted Binary Tree Example Insert in an empty Tree the elements 5, 1, 3, 2, 8, 4, 6, 7 and 9: 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 37 5 1 3 2 4 9 8 7 6
- 38. Delete a node in a Sorted Binary Tree Be u ϵ T the node to be removed from Tree T Be U the sub tree from T with root(U) = u If the left Sub Tree U1 of U is empty than replace U by U1 If the right Sub Tree U2 of U is empty than replace U by U2 If both Sub Trees are not empty than: Chose an u2 ϵ U2 such that v(u2) ≤ v(t) for all t ϵ U2 u2 has to be removed from U2 u2 replace the node u The effort to delete a node in a Tree is O(h(T)) due to the search for the minimal element u2 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 38
- 39. Delete a node in a Sorted Binary Tree Example 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 39
- 40. Other Tree Types 2-3 tree 2-3-4 tree AA tree AVL tree B-tree Elastic binary tree Random binary tree Red-black tree Self-balancing binary search tree Unrooted binary tree 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 40
- 41. Any questions? 23/10/2018 Lecture 4 Dynamic Data Structure Part 2 41