I2206 Data Structures TS 6: Binary Trees - Binary Search Trees

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1 Lebanese University BS - Computer Science Faculty of Science Section I I2206 Data Structures TS 6: Binary Trees - Binary Search Trees Exercise 1 What value does the following C function return? i n t mystery ( Btree x ) i f ( x == NULL) r eturn 0 ; e l s e r eturn max( mystery ( x >l e f t ), mystery ( x >r i g h t ) ) ; } Exercise 2 Write a function that returns the maximum value of all the nodes in a binary tree. Assume all values are nonnegative; return -1 if the tree is empty. Exercise 3 Write a function that prints all the nodes less than a given value v in a binary tree of integers. Exercise 4 Write a function that returns the sum of all the nodes in a binary tree. Exercise 5 Write a function that counts the number of items in a binary tree. Exercise 6 Write an iterative function that tests whether an element belongs to a BST. Exercise 7 Write a recursive function that tests whether an element belongs to a BST. Exercise 8 Write a recursive function that inserts an element into a BST. Exercise 9 Write an iterative function that inserts an element into a BST. Exercise 10 Write a function that checks whether two BSTs are equal (contains the same elements). Exercise 11 Write a function that deletes an element from a BST. Exercise 12 A binary tree is said to be balanced if both of its subtrees are balanced and the height of its left subtree differs from the height of its right subtree by at most 1. Write a C function to determine whether a given binary tree is balanced. Exercise 13 Write a function that returns the depth of a binary tree. Test your function. Exercise 14 Write a function that counts the number of leaf nodes in a binary tree. Test your function. 1

2 Exercise 15 Write a function that counts the number of non-leaf nodes in a binary tree. Test your function. Exercise 16 Write a function that computes the maximum width of a dynamically implemented BT. We define the maximum width of a BT as the maximum number of nodes located at the same level. Exercise 17 A comb left is a locally complete binary tree (each node has 0 or 2 children) in which each right child of a node is a leaf. Write a function isleftcomb that checks if a binary tree is a left comb. Exercise 18 Write a function that computes the maximum depth of a statically implemented BT. Exercise 19 ( Exam) A historian wants to classify a set of events. Each event is characterized by a title, a location and a period of time (as most of the events are spread over several years). Title and location are strings, while the period is a closed interval [start, end], where start and end is a pair of integers such that < start end < +. In the following, we use the following declarations: typedef struct int start, end;} interval; typedef struct char *title, *location; interval period;} Event; Here are some examples of events: Title Location Period World War II world [ ] First Atomic Bomb Hiroshima [ ] JFK president U SA [ ] Independence Lebanon [ ] Note that certain events are spread over several years, while others begin and end the same year. In order to classify these events, the historian builds a binary tree of events (BtreeE) whose data are events such as: If EvL is in the left subtree of Ev, then EvL.start Ev.start If EvR is in the right subtree of Ev, then EvR.start > Ev.start Thus, an event tree is a binary search tree for the event starting year in which repetitions can occur since EvL.start Ev.start (see figure against, where T i denotes a title and L i a location). We therefore have the following declaration: typedef struct node Event data; struct node *Left, *Right;} BtreeE; 2

3 1. Write a recursive function int contains(btreee B, int year) that returns the number of events of B such as year interval of these events. 2. Write an iterative function that returns the minimum interval that contains all the intervals of events of a given BtreeE. In the previous example, the result will be [1210, 1985]. 3. Write a function int isbtreee(btreee B) that determines whether a given event tree B satisfies the properties of a BtreeE. 4. Write a function int LongestInterval(BtreeE B, Event E) that returns the time in years from the largest inersection event between E and those of B. Example: the intersection of the event T8, L8, [ ] with the previous tree is 31; it comes from the intersection with T2, L2, [ ]. 5. Write a function int maxlong(btreee B) that returns the longest common period (intersection) of two events in a given BtreeE. In the previous tree, the events, T2,L2,[ ] and T5,L5,[ ] have an intersection of 24 years (between 1930 and 1954). Exercise 20 In a binary tree (BT), a path between the root and a leaf is the number of arcs joining the root to the leaf. The BT height is the length of the longest path between the root and each of the leaves. 1. Write a function int height(bt T) that returns the height of T. 2. The search algorithm in a BST is proportional to its height. In order to improve the search performance, we consider the class of balanced BST. A BST T is balanced if T is empty or if for each node, the difference between the heights of the left and right subtrees is < 2. Write a function int dif LR(BT T) that returns the difference between left and right subtree height of a given BT T. 3. Write a function int balanced(bt T) that returns 1 if T is balanced and 0 otherwise. 4. In order to ensure the balance of a BST, we suggest to transform it using the following rotations: Left Rotation: a left rotation around the node B consists in getting down B and getting up its right child D without invalidating elements order (see figure below). Right Rotation: the reverse of the left rotation. Left-Right Rotation on T: it consists in a left rotation on T left subtree followed by a right rotation on T. Right-Left Rotation on T: it consists in a right rotation on T right subtree followed by a left rotation on T. 3

4 Write a function left rotation that rotates left a given BST. You must only manipulate pointers; no memory allocation or release is allowed. 5. Write a function left right rotation that rotates left-right a given BST. 6. The insertion of an element into a balanced BT is as follows: the element is added as when inserting into a BST then: If the left balance is lost then a left-right rotation is done if the left child lean to the right, otherwise a simple right rotation is done. If the right balance is lost then a right-left rotation is done if the right child lean to the left, otherwise a simple left rotation is done. Assuming that a binary tree T is said to be left-leaning if dif LR(T)>0, and suppose having the functions: lean L, lean R, left rotation, left right rotation, right rotation, and right left rotation, write a function to insert an element into a balanced BST that implements the described insertion procedure. Exercise 21 ( Exam) 1. (10 pts) Write a recursive function int max rec(btree B, int root index) that returns the maximum element in a binary tree statically implemented rooted at index root index. Example: Consider the following Binary tree and its representation using an array: Index Root Left Right 0 Rnd Rnd Rnd Rnd -1 Rnd where Rnd designates a random number. max rec(b, 1) returns 45; max rec(b, 2) returns 4 since the maximum element of the tree rooted at element of value 4 (root index=2) is equal to 4; 4

5 max rec(b, 5) returns (2 pts) Without performing calculation, give the complexity of the above function. 3. (11 pts) Write a recursive function int isbst(btree tree) that checks whether a statically implemented Binary tree is a Binary Search Tree. Your function should return 0 for the tree above. 4. (10 pts) Write an iterative function that calculates the width of a statically implemented Binary tree. Recall that the width of a tree is equal to the maximum of widths of all levels. Your function should return 3 for the tree above. Exercise 22 ( Exam) We define the maximum width of a BT as the maximum number of nodes located at the same level, and the maximum depth as the longest path from the root to a leaf. 1. Write a function that computes the maximum width of a dynamically implemented BT. 2. Write a function that computes the maximum depth of a statically implemented BT. Exercise 23 ( Exam) A binary tree is called complete if all its non-leaf nodes have two children. Write a function that tests whether a binary tree statically implemented is complete. Exercise 24 ( Exam) Write a function deepest(a, E) that returns the deepest level of the element E in the binary tree A or 0 if E / A. We assume that the root of the tree is at level 1, and that the tree is dynamically implemented. Running example: let A be the following BT; deepest(a,2)=3, deepest(a,3)= Exercise 25 ( Exam) We define the width of a level of a BT as the number of nodes located at this level. A BT is a fairly well developed if the width of each level of the BT is > than the one of the previous level. Write a function that determines whether a BT is fairly well developed Exercise 26 ( Exam) Consider the following declaration of a BST in which the field nbd denotes the number of descendants of a tree node: typedef s t r u c t node element data ; 5

6 } BST; i n t nbd ; s t r u c t node Left, Right ; 1. Write a function that inserts an element to a BST; 2. Write a function that tests that the field nbd of each BST node is correct. Exercise 27 ( Exam) A Boolean expression is represented as a binary tree of strings (BT) in which internal nodes contain the logical connectors AND, OR and NOT, and leaves contain strings representing Boolean variables. Ex: the expression a 1 AND NOT (bool 2 OR bx 30 ) is represented by the following BT: An interpretation of an expression is represented by a set of pairs <variable, value> that associates to each variable of the expression a Boolean value. Ex: the set (<a1,1>,<bool2,0>,<bx30,1>) associate the value 1 (True) to the Boolean variables a1 and bx30 and the value 0 (False) to bool Assume that we use BT dynamic implementation; give the declaration of the types Interpretation and Expression. 2. Write a function that returns the Boolean value of a variable in an interpretation if it exists. 3. Write a function int Evaluate(Interpretation I, Expression E, int *Value) that computes the value of the expression E according to the interpretation I. 4. Now, we want to test whether a given expression is valid (true for all interpretations). Assume the existence of an array containing all interpretations of a given expression (all possible variables combinations), write a function int Valid(Interpretation arrayinter[], int NBInter, Expression E) that checks the validity of the expression. Exercise 28 Write a method and the corresponding recursive function to insert an Entry, passed as a parameter, into a binary tree. If the root is empty, the new entry should be inserted into the root, otherwise it should be inserted into the shorter of the two subtrees of the root (or into the left subtree if both subtrees have the same height). Exercise Suppose that Entry is the type char. Write a function that will print all the entries from a binary tree in the bracketed form (data: LT, RT) where data is the Entry in the root, LT denotes the left subtree of the root printed in bracketed form, and RT denotes the right subtree in bracketed form. For example, the first tree in Figure 10.3 will be printed as (. : (a: (:, ), (:, )), (b: (:, ), (:, ))) 2. Modify the function so that it prints nothing instead of (:, ) for an empty subtree, and x instead of (x:, ) for a subtree consisting of only one node with the Entry x. Hence the preceding tree will now print as (. : a, b). 6

7 Exercise 30 Write a function that will interchange all left and right subtrees in a linked binary tree. Exercise 31 Consider a BST of integers. 1. Write a function that returns the number of common entries in two BST. 2. Write a function that merges 2 BST into a new BST. Exercise We consider a variant of BST where left(n) n < right(n) and this for any node n in the tree. Write a function that calculates the number of distinct values in this kind of binary tree. Give the corresponding declaration. 2. Given 2 BST A and B, write a function that deletes from B all the common elements with A. Exercise 33 A Binary Search Tree (BST) of whole integers can be represented by an array where the root is located at 0 and the contents of the remaining nodes correspond to the course in width of the tree. i.e. every node has two locations for its left and right children. In the case where one (or both) of the two children doesn t exist, the corresponding location will contain the value -1. Ex : The array corresponds to the tree 1. Write the corresponding declaration. 2. Write a function that inserts a natural integer in a BST. 3. Write a function that calculates the number of nodes having only one child in a BST. 4. Write a function that tests if a BT is a Binary Search Tree. Exercise 34 Consider a binary tree T. We define the path from the root to a leaf as the number of edges from the root to that leaf. The minimum path in a tree T is the shortest path among all paths between the root and all leaves. 1. Write a function int min(t) that returns the minimum path in the tree T. 2. Write a function int diff(t) that returns the difference between the minimum and maximum paths in the tree T. 3. A binary tree T is said to be leftist if, for each node N in T, diff(left(n)) diff(right(n)). Write a function int leftest(t) that determines if T is leftist or not. 7

8 Exercise Write a function that returns the value of the father of a given element in a binary tree of integers without repetition. 2. Write a function that returns an array containing the values of all ancestors of a given element in a binary tree of integers without repetition ordered from the nearest to the farthest. Example: the sorted ancestors of 14 in the following tree are: 9, 17, Write a function that returns the value of the nearest common ancestor of two given elements in a binary tree of integers without repetition. Example: the nearest common ancestor of 8 and 5 is 17. Exercise 36 Write a function that displays the elements of a binary tree located at odd left to right and top to bottom. The root of the tree is assumed to be at level 0. Example: Expected result: B C H I J K Exercise 37 A comb left is a locally complete binary tree (each node has 0 or 2 children) in which each right child of a node is a leaf. Write a function isleftcomb that checks if a binary tree is a left comb. Exercise 38 Consider a binary tree of integers dynamically implemented; write a function update that changes node values such that each node is equal to the sum of its children. It is obvious that leaves remain 8

9 unchanged. Example: Exercise 39 A complete binary tree is a binary tree in which all level are completely filled: each non-leaf node has exactly 2 children and all leaves are located in the same level. Thus a tree of depth n has 2n-1 nodes ( ). In this exercise, we represent a complete binary tree statically by an array that corresponds to the tree in width traversal (the root is at the index 0). In this exercise, we suppose that all tree values are distinct. Ex: the below array corresponds to the following BT: We define the ancestors of a leaf L as all nodes that are on the path from L to the root (including L). Thus, in this example, the ancestors of 18 are 18, 5, 4 and Write a function that takes a leaf L in parameter and computes the sum of L ancestors. 2. Write a function that takes in parameters 2 leaves and computes the sum of their ancestors without repetition. Ex: for the leaves 7 and 18, the sum of ancestors is: (4 and 6 are taken into account only once). Exercise 40 ( Exam) A Trees Traversal Consider the following tree:

10 1. (2 pts) Give the inorder traversal of the above tree. 2. (2 pts) Write a recursive function that prints the inorder traversal of a tree dynamically implemented. 3. (10 pts) Write an iterative function that prints the inorder traversal of a tree dynamically implemented. B BST The following function checks whether a given binary tree dynamically implemented is a BST or not. i n t isbst ( Btree t r e e ) i f ( t r e e == NULL) return 1 ; i f ( tree >l e f t!= NULL && FindMax ( tree >l e f t ) > tree >data ) return 0 ; i f ( tree >r i g h t!= NULL && FindMin ( tree >r i g h t ) < tree >data ) return 0 i f (! isbst ( tree >l e f t )! isbst ( tree >r i g h t ) ) return 0 ; r eturn 1 ; } FindMax and FindMin are 2 auxiliary functions that returns the maximum (resp. minimum) element of a BST. element FindMin ( Btree t r e e ) i f ( t r e e==null) return INT MAX; r eturn min ( tree >data, min ( FindMin ( tree >l e f t ), FindMin ( tree >r i g h t ) ) ) ; } element FindMax ( Btree t r e e ) i f ( t r e e==null) r eturn INT MIN ; r eturn max( tree >data, max( FindMax ( tree >l e f t ), FindMax ( tree >r i g h t ) ) ) ; } 1. (1 pt) Without performing calculation, give the worst-case time complexity of the isbst function. Justify your answer. 2. (6 pts) Note that the inorder traversal of BSTs produces sorted lists. While traversing the BST in inorder, and at each node, check the condition that its key value should be greater then the key value of its previous visited node. Write a recursive function that checks whether a given binary tree is a BST or not given the above approach. 3. (1 pt) Without performing calculation, give the worst-case time complexity of your function written in part B-2. Justify your answer. C Max-Heap We are interested in this part to implement a stack of integers using a Max-Heap. Let H be a Max-Heap allowing the following operations: Max-Heap CreateHeap(); This operation creates a Max-Heap. The capacity of the heap is fixed and cannot increase. int insert(max-heap *H, element e); This operation inserts an element e in the Max-Heap if there s room left. The element e could be a complex type and percolate up could be performed on one of its field members. int deletemax(max-heap *H); This operation deletes the maximum element in a Max-Heap if it exists. 10

11 int getmax(max-heap H, element *e); This operation returns the Max-Heap maximum element in e if it exists. int isemptyheap(max-heap H); This operation returns true if the Max-Heap is empty, false otherwise. int isfullheap(max-heap H); This operation returns true if the Max-Heap is full, false otherwise. 1. (2 pts) Give the declaration of types element and stack; 2. (8 pts) Implement a stack using a Max-Heap. D Generic Trees A Generic Tree can be represented as follows: Element First Child Next Sibling A NULL NULL B F L NULL... D NULL NULL The tree node declaration for general trees can be given as: typedef... element ; typedef s t r u c t node element data ; s t r u c t node f i r s t C h i l d ; s t r u c t node n e x t S i b l i n g ; } GenericTree ; 1. (3 pts) Write a function that returns the number of nodes in a Generic Tree. 11