1.遞歸函數
在函數內部，可以調用其他函數。如果一個函數在內部調用自身本身，這個函數就是遞歸函數。
舉個例子，我們來計算階乘n! = 1 x 2 x 3 x ... x n，用函數fact(n)表示，可以看出：
fact(n) = n! = 1 x 2 x 3 x ... x (n-1) x n = (n-1)! x n = fact(n-1) x n
所以，fact(n)可以表示為n x fact(n-1)，只有n=1時需要特殊處理。
於是，fact(n)用遞歸的方式寫出來就是：
def fact(n): if n==1: return 1 return n * fact(n - 1)
上面就是一個遞歸函數。可以試試：
>>> fact(1) 1 >>> fact(5) 120 >>> fact(100) 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000L
如果我們計算fact(5)，可以根據函數定義看到計算過程如下：
===> fact(5) ===> 5 * fact(4) ===> 5 * (4 * fact(3)) ===> 5 * (4 * (3 * fact(2))) ===> 5 * (4 * (3 * (2 * fact(1)))) ===> 5 * (4 * (3 * (2 * 1))) ===> 5 * (4 * (3 * 2)) ===> 5 * (4 * 6) ===> 5 * 24 ===> 120
遞歸函數的優點是定義簡單，邏輯清晰。理論上，所有的遞歸函數都可以寫成循環的方式，但循環的邏輯不如遞歸清晰。
使用遞歸函數需要注意防止棧溢出。
2.棧溢出
在計算機中，函數調用是通過棧（stack）這種數據結構實現的，每當進入一個函數調用，棧就會加一層棧幀，每當函數返回，棧就會減一層棧幀。由於棧的大小不是無限的，所以，遞歸調用的次數過多，會導致棧溢出。可以試試fact(1000)：
>>> fact(1000) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "<stdin>", line 4, in fact ... File "<stdin>", line 4, in fact RuntimeError: maximum recursion depth exceeded
尾遞歸
解決遞歸調用棧溢出的方法是通過尾遞歸優化，事實上尾遞歸和循環的效果是一樣的，所以，把循環看成是一種特殊的尾遞歸函數也是可以的。
尾遞歸是指，在函數返回的時候，調用自身本身，並且，return語句不能包含表達式。這樣，編譯器或者解釋器就可以把尾遞歸做優化，使遞歸本身無論調用多少次，都只占用一個棧幀，不會出現棧溢出的情況。
上面的fact(n)函數由於return n * fact(n - 1)引入了乘法表達式，所以就不是尾遞歸了。要改成尾遞歸方式，需要多一點代碼，主要是要把每一步的乘積傳入到遞歸函數中：
def fact(n): return fact_iter(1, 1, n) def fact_iter(product, count, max): if count > max: return product return fact_iter(product * count, count + 1, max)
可以看到，return fact_iter(product * count, count + 1, max)僅返回遞歸函數本身，product * count和count + 1在函數調用前就會被計算，不影響函數調用。
fact(5)對應的fact_iter(1, 1, 5)的調用如下：
===> fact_iter(1, 1, 5) ===> fact_iter(1, 2, 5) ===> fact_iter(2, 3, 5) ===> fact_iter(6, 4, 5) ===> fact_iter(24, 5, 5) ===> fact_iter(120, 6, 5) ===> 120
尾遞歸調用時，如果做了優化，棧不會增長，因此，無論多少次調用也不會導致棧溢出。
遺憾的是，大多數編程語言沒有針對尾遞歸做優化，Python解釋器也沒有做優化，所以，即使把上面的fact(n)函數改成尾遞歸方式，也會導致棧溢出。
優化尾遞歸的裝飾器
有一個針對尾遞歸優化的decorator，可以參考源碼：
#!/usr/bin/env python2.4 # This program shows off a python decorator( # which implements tail call optimization. It # does this by throwing an exception if it is # it's own grandparent, and catching such # exceptions to recall the stack. import sys class TailRecurseException: def __init__(self, args, kwargs): self.args = args self.kwargs = kwargs def tail_call_optimized(g): """ This function decorates a function with tail call optimization. It does this by throwing an exception if it is it's own grandparent, and catching such exceptions to fake the tail call optimization. This function fails if the decorated function recurses in a non-tail context. """ def func(*args, **kwargs): f = sys._getframe() if f.f_back and f.f_back.f_back \ and f.f_back.f_back.f_code == f.f_code: raise TailRecurseException(args, kwargs) else: while 1: try: return g(*args, **kwargs) except TailRecurseException, e: args = e.args kwargs = e.kwargs func.__doc__ = g.__doc__ return func @tail_call_optimized def factorial(n, acc=1): "calculate a factorial" if n == 0: return acc return factorial(n-1, n*acc) print factorial(10000) # prints a big, big number, # but doesn't hit the recursion limit. @tail_call_optimized def fib(i, current = 0, next = 1): if i == 0: return current else: return fib(i - 1, next, current + next) print fib(10000) # also prints a big number, # but doesn't hit the recursion limit.
現在，只需要使用這個@tail_call_optimized，就可以順利計算出fact(1000)：
>>> fact(1000) 4023872600770937735437024339230039857193748642107146325437999104299385123986290205920442084869694048
0047998861019719605863166687299480855890132382966994459099742450408707375991882362772718873251977950
5950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476
6328898359557354325131853239584630755574091142624174743493475534286465766116677973966688202912073791
4385371958824980812686783837455973174613608537953452422158659320192809087829730843139284440328123155
8611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151
0273418279777047846358681701643650241536913982812648102130927612448963599287051149649754199093422215
6683257208082133318611681155361583654698404670897560290095053761647584772842188967964624494516076535
3408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200
0158885351473316117021039681759215109077880193931781141945452572238655414610628921879602238389714760
8850627686296714667469756291123408243920816015378088989396451826324367161676217916890977991190375403
1274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786
9061172601587835207515162842255402651704833042261439742869330616908979684825901254583271682264580665
2676995865268227280707578139185817888965220816434834482599326604336766017699961283186078838615027946
5955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657
2450144028218852524709351906209290231364932734975655139587205596542287497740114133469627154228458623
7738753823048386568897646192738381490014076731044664025989949022222176590433990188601856652648506179
9702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819
3723886426148396573822911231250241866493531439701374285319266498753372189406942814341185201580141233
4482801505139969429015348307764456909907315243327828826986460278986432113908350621709500259738986355
4277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994
8717012445164612603790293091208890869420285106401821543994571568059418727489980942547421735824010636
7740459574178516082923013535808184009699637252423056085590370062427124341690900415369010593398383577
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