CCC '18 S4 - Balanced Trees - DMOJ: Modern Online Judge

CCC '18 S4 - Balanced Trees

Points: 15 (partial)

Time limit: 1.0s

Memory limit: 256M

Author:

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

In particular, all $k$ ~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 $k$ ~k~ subtrees does not exceed $w$ ~w~, the weight of the overall tree. For example, if a perfectly balanced tree of weight $8$ ~8~ has $3$ ~3~ subtrees, then each subtree would have weight $2$ ~2~, since $2+2+2 = 6 \le 8$ ~2+2+2 = 6 \le 8~.

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

Input Specification

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

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

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

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

Output Specification

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

Sample Input 1

Sample Output 1

Explanation for Sample Output 1

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

Sample Input 2

Sample Output 2

Comments

13

elliss commented on Feb. 4, 2019, 4:25 p.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

2

RAiNcalleER commented on Feb. 3, 2022, 10:07 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. 28, 2019, 3:43 a.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..

13

Plasmatic commented on Jan. 28, 2019, 4:02 a.m.

You're running out of memory