Tree traversals. Review: recursion Tree traversals. October 05, 2017 Cinda Heeren / Geoffrey Tien 1

Transcription

1 Tree traversals Review: recursion Tree traversals Cinda Heeren / Geoffrey Tien 1

2 Rabbits! What happens when you put a pair of rabbits in a field? More rabbits! Let s model the rabbit population, with a few assumptions: Newly-born rabbits take one month to reach maturity and mate Each pair of rabbits produces another pair of rabbits one month after mating Rabbits never die...and recursion Cinda Heeren / Geoffrey Tien 2

3 More rabbits... How many rabbit pairs are there after 5 months? Month 1: start 1 pair Month 2: first pair are now mature and mate 1 pair Month 3: first pair give birth to a pair of babies original pair + baby pair = 2 pairs Month 4: first pair give birth to another pair of babies, pair born in month 3 are now mature 3 pairs Month Month 5: the 3 pairs from month 4, and two new pairs 5 pairs Month 6: the 5 pairs from month 5, and three new pairs 8 pairs And so on... Cinda Heeren / Geoffrey Tien 3

4 Fibonacci series The n th number in the Fibonacci series, fib(n), is: 0 if n = 0, and 1 if n = 1 fib(n 1) + fib(n 2) for any n > 1 e.g. what is fib(23) Easy if we only knew fib(22) and fib(21) The answer is fib(22) + fib(21) What happens if we actually write a function to calculate Fibonacci numbers like this? Cinda Heeren / Geoffrey Tien 4

5 Calculating the Fibonacci series Let s write a function just like the formula fib(n) = 0 if n = 0, 1 if n = 1, otherwise fib(n) = fib(n 1) + fib(n 2) int fib(int n) { if (n <= 1) return max(0, n); else return fib(n-1) + fib(n-2); } The function calls itself Cinda Heeren / Geoffrey Tien 5

6 Recursive functions The Fibonacci function is recursive A recursive function calls itself Each call to a recursive method results in a separate call to the method, with its own input Recursive functions are just like other functions The invocation (e.g. parameters, etc.) is pushed onto the call stack And removed from the call stack when the end of a method or a return statement is reached Execution returns to the previous method call Cinda Heeren / Geoffrey Tien 6

7 Recursive function anatomy Recursive functions do not use loops to repeat instructions But use recursive calls, in if statements Recursive functions consist of two or more cases, there must be at least one Base case, and Recursive case Cinda Heeren / Geoffrey Tien 7

8 Recursion cases The base case is a smaller problem with a known solution This problem s solution must not be recursive Otherwise the function may never terminate There can be more than one base case And base cases may be implicit The recursive case is the same problem with smaller input The recursive case must include a recursive function call There can be more than one recursive case Cinda Heeren / Geoffrey Tien 8

9 Analysis of fib(5) int fib(int n) { if (n <= 1) return max(0, n); else return fib(n-1) + fib(n-2); } 3 5 fib(5) 2 fib(4) fib(3) fib(3) fib(2) fib(2) fib(1) fib(2) fib(1) fib(1) fib(0) fib(1) fib(0) 1 fib(1) 0 fib(0) Later in the course we will explain how this is an extremely inefficient way to compute the Fibonacci series Cinda Heeren / Geoffrey Tien 9

10 Example Base cases Target is found, or the end of the array is reached Recursive case Recursive linear search Target not found, search subarray starting from next element // Recursive linear search int reclinsearch(int arr[], int next, int sz, int x) { if (next >= sz) // end of array reached return -1; else if (x == arr[next]) // target found return next; else // not found, search from different starting index return reclinsearch(arr, next + 1, sz, x); } Cinda Heeren / Geoffrey Tien 10

11 Back on Topic binary tree traversal A traversal algorithm for a binary tree s each node in the tree Typically, it will do something while ing each node! Traversal algorithms are naturally recursive There are three traversal methods inorder preorder postorder Cinda Heeren / Geoffrey Tien 11

12 inorder traversal algorithm void inorder(node* nd) { if (nd!= NULL) { inorder(nd->leftchild); (nd); inorder(nd->rightchild); } } The function would do whatever the purpose of the traversal is (e.g. print the data value of the node). Cinda Heeren / Geoffrey Tien 12

13 preorder Traversal (nd); preorder(nd->leftchild); preorder(nd->rightchild); preorder(left) preorder(right) 6 27 preorder(left) preorder(right) 3 9 preorder(left) preorder(right) preorder(left) preorder(right) preorder(left) 39 preorder(right) preorder(left) preorder(right) preorder(left) preorder(right) 11 Cinda Heeren / Geoffrey Tien 13

14 postorder traversal postorder(nd->leftchild); postorder(nd->rightchild); (nd); postorder(left) postorder(right) 7 27 postorder(left) postorder(right) 2 9 postorder(left) postorder(right) postorder(left) postorder(right) postorder(left) postorder(right) 5 6 postorder(left) 20 postorder(right) 39 postorder(left) postorder(right) Cinda Heeren / Geoffrey Tien 14

15 Exercise What will be printed by an in-order traversal of the tree? preorder? postorder? inorder(nd->leftchild); (nd); inorder(nd->rightchild); Food for thought: which traversal will be useful for deep copy? Deep delete? 39 Cinda Heeren / Geoffrey Tien 15

16 Readings for this lesson Koffman Portions of Chapter (for review) Chapter (Tree traversals and binary trees) Cinda Heeren / Geoffrey Tien 16