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Tree Recursion Tree Recursion Tree-shaped processes arise whenever executing the body of a recursive function makes more than one recursive call n: 0, 1, 2, 3, 4, 5, 6, 7, 8, ... , 35 fib(n): 0, 1, 1, 2, 3, 5, 8, 13, 21, ... ,

  1. Tree Recursion

  2. Tree Recursion Tree-shaped processes arise whenever executing the body of a recursive function makes more than one recursive call n: 0, 1, 2, 3, 4, 5, 6, 7, 8, ... , 35 fib(n): 0, 1, 1, 2, 3, 5, 8, 13, 21, ... , 9,227,465 def fib (n): if n == 0 : return 0 elif n == 1 : return 1 else : return fib(n- 2 ) + fib(n- 1 ) ! 9 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  3. A Tree-Recursive Process The computational process of fib evolves into a tree structure fib(5) fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 (Demo) ! 10

  4. Repetition in Tree-Recursive Computation This process is highly repetitive; fib is called on the same argument multiple times fib(5) fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 (We will speed up this computation dramatically in a few weeks by remembering results) ! 11

  5. g a blow to a c't I VII r f 1 or n m I return 1 n 17 ng g path m m l return path M tofokwa y µ of ways to get to cm Total t N M l of ways to get to Total N l of ways to get to l M Total took I step towards N

  6. of ways to get to l M N Total t N M l of ways to get to Total N l of ways to get to M Total paths 2,2 T stet w f Z 1 z path CZ 1 Z Path C l t 1 2

  8. Example: Counting Partitions

  9. Counting Partitions The number of partitions of a positive integer n, using parts up to size m, is the number of ways in which n can be expressed as the sum of positive integer parts up to m in increasing order. count_partitions(6, 4) 2 + 4 = 6 1 + 1 + 4 = 6 3 + 3 = 6 1 + 2 + 3 = 6 1 + 1 + 1 + 3 = 6 2 + 2 + 2 = 6 1 + 1 + 2 + 2 = 6 1 + 1 + 1 + 1 + 2 = 6 1 + 1 + 1 + 1 + 1 + 1 = 6 ! 13

  10. Counting Partitions The number of partitions of a positive integer n, using parts up to size m, is the number of ways in which n can be expressed as the sum of positive integer parts up to m in non- decreasing order. count_partitions(6, 4) • Recursive decomposition: finding simpler instances of the problem. • Explore two possibilities: • Use at least one 4 • Don't use any 4 • Solve two simpler problems: • count_partitions(2, 4) • count_partitions(6, 3) • Tree recursion often involves exploring different choices. ! 14

  11. Counting Partitions The number of partitions of a positive integer n, using parts up to size m, is the number of ways in which n can be expressed as the sum of positive integer parts up to m in increasing order. def count_partitions(n, m): • Recursive decomposition: finding if n == 0: simpler instances of the problem. return 1 • Explore two possibilities: elif n < 0: return 0 • Use at least one 4 elif m == 0: • Don't use any 4 return 0 • Solve two simpler problems: else: with_m = count_partitions(n-m, m) • count_partitions(2, 4) without_m = count_partitions(n, m-1) • count_partitions(6, 3) return with_m + without_m • Tree recursion often involves exploring different choices. (Demo) ! 15 pythontutor.com/composingprograms.html#code=def%20count_partitions%28n,%20m%29%3A%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20elif%20n%20<%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20elif%20m%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20with_m%20%3D%20count_partitions%28n-m, %20m%29%20%0A%20%20%20%20%20%20%20%20without_m%20%3D%20count_partitions%28n, %20m-1%29%0A%20%20%20%20%20%20%20%20return%20with_m%20%2B%20without_m%0A%20%20%20%20%20%20%20%20%0Aresult%20%3D%20count_partitions%285,%203%29%0A%0A#%201%20%2B%201%20%2B%201%20%2B%201%20%2B%201%20%3D%205%0A#%201%20%2B%201%20%2B%201%20%2B%202%20%2B%20%20%20%3D%205%0A#%201%20%2B%202%20%2B%202%20%2B%20%20%20%20%20%20%20%3D%205%0A#%201%20%2B%201%20%2B%203%20%2B%20%20%20%20%20%20%20%3D%205%0A#%202% 20%2B%203%20%2B%20%20%20%20%20%20%20%20%20%20%20%3D%205&mode=display&origin=composingprograms.js&cumulative=false&py=3&rawInputLstJSON=[]&curInstr=0

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