问题：

任何数都能分解成2的幂，比如

7=1+1+1+1+1+1+1

=1+1+1+1+1+2

=1+1+1+2+2

=1+2+2+2

=1+1+1+4

=1+2+4

共有6种分解方式，设f(n)为任意正整数可能分解总数，比如f(7)=6

写个算法，输入数，求出其分解的总数。

思路：

先按照树形结构，把一个数可能的2的幂的子数记录下来，比如7拆分成7个1，3个2，1个4。从高到底遍历所有可能的搭配。

import math

import copy

def get_distribute_for_number(number):

distribute = {}

distribute[0] = number

for i in range(0, int(math.log(number, 2) + 1)):

if i not in distribute:

break

count_i = distribute[i]

while count_i >= 2:

count_i -= 2

if i + 1 not in distribute:

distribute[i + 1] = 0

distribute[i + 1] += 1

return distribute

def count_by_distribute(distribute, number, parent_expr):

if number == 0:

print("expr : %s"%parent_expr[:-3])

return 1

max_leaf = len(distribute) - 1

if max_leaf == 0:

print("expr : %s"%(parent_expr + " 1 * %d"%number))

return 1

curr_distribute = copy.copy(distribute)

max_leaf_value = 2 ** max_leaf

max_leaf_num = curr_distribute.pop(max_leaf)

count = 0

for i in range(0, max_leaf_num + 1):

left = number - max_leaf_value * i

expr = parent_expr

expr += "%d * %d"%(max_leaf_value, i)

expr += " + "

if left < 0:

break

count_left = count_by_distribute(curr_distribute, left, expr)

count += count_left

#print("current distribute is ")

#print(curr_distribute)

#print("kept %d leaves of %d"%(i, max_leaf_value))

#print("distributing num for %d is %d"%(left, count_left))

return count

number = input("Input Number:")

distribute = get_distribute_for_number(number)

print("Distribute for num is")

print(distribute)

count = count_by_distribute(distribute, number, "")

print("Number %d can be distributed by %d ways"%(number, count))