CCC '18 S4 - Balanced Trees

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Points: 15 (partial)
Time limit: 1.0s
Memory limit: 256M

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Problem type
Canadian Computing Competition: 2018 Stage 1, Senior #4

Trees have many fascinating properties. While this is primarily true for trees in nature, the concept of trees in math and computer science is also interesting. A particular kind of tree, a perfectly balanced tree, is defined as follows.

Every perfectly balanced tree has a positive integer weight. A perfectly balanced tree of weight 1 always consists of a single node. Otherwise, if the weight of a perfectly balanced tree is w and w \ge 2, then the tree consists of a root node with branches to k subtrees, such that 2 \le k \le w. In this case, all k subtrees must be completely identical, and be perfectly balanced themselves.

In particular, all k subtrees must have the same weight. This common weight must be the maximum integer value such that the sum of the weights of all k subtrees does not exceed w, the weight of the overall tree. For example, if a perfectly balanced tree of weight 8 has 3 subtrees, then each subtree would have weight 2, since 2+2+2 = 6 \le 8.

Given N, find the number of perfectly balanced trees with weight N.

Input Specification

The input will be a single line containing the integer N (1 \le N \le 10^9).

For 5 of the 15 marks available, N \le 1\,000.

For an additional 2 of the 15 marks available, N \le 50\,000.

For an additional 2 of the 15 marks available, N \le 10^6.

Output Specification

Output a single integer, the number of perfectly balanced trees with weight N.

Sample Input 1

4

Sample Output 1

3

Explanation for Sample Output 1

One tree has a root with four subtrees of weight 1; a second tree has a root with two subtrees of weight 2; the third tree has a root with three subtrees of weight 1.

Sample Input 2

10

Sample Output 2

13

Comments


  • 11
    elliss  commented on Feb. 4, 2019, 11:25 a.m.

    I still have questions about the second sample input. I tried and got the same result as that guy got--17. Can anyone specify what the situation is when input is 10? ty


    • 0
      RAiNcalleER  commented on Feb. 3, 2022, 5:07 p.m.

      For all trees with a weight of 10 and # of branches in the range of [4,10], we +1 each time to the total number of perfect trees.

      With 3 branches (each branch will have a weight of 10/3=3), we add 2 to our answer(since 3->1,1 and 3->1,1,1).

      With 2 branches (each branch will have a weight of 10/2=5), we add 4 to our answer(since 5->1,1,1,1,1; 5->1,1,1,1; 5->1,1,1; 5->2,2).


  • 0
    Summertony717  commented on Jan. 27, 2019, 10:43 p.m.

    I don't know why I get seg fault on the last batch... I doubled checked my indexing, and nothing seems to be wrong..


    • 8
      Plasmatic  commented on Jan. 27, 2019, 11:02 p.m.

      You're running out of memory