시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 20 16 15 78.947%

문제

다음과 과정을 거쳐서 바이너리 트리에 번호를 붙일 수 있다.

  • 비어있는 트리의 번호는 0이다.
  • 노드가 1개인 트리는 1번이다.
  • 노드가 m개인 바이너리 트리의 번호는 m+1개인 트리의 번호보다 작다.
  • 노드가 m개이고, 왼쪽과 오른쪽 서브트리가 각각 L과 R인 바이너리 트리의 번호 n은 아래와 같은 두 조건 중 하나라도 만족하는 트리보다 번호가 작다. 
    • 왼쪽 서브 트리가 L보다 번호가 크다. 또는
    • 왼쪽 서브 트리가 L과 같고, 오른쪽 서브트리가 R보다 번호가 크다.

처음 10개 바이너리 트리와 20번째 바이너리 트리를 그림으로 그려보면 아래와 같다.

        0  1  2      3  4      5      6      7      8  9        ...     20

           X  X      X  X      X      X      X      X  X                 X
               \    /    \      \    / \    /      /    \               /
                X  X      X      X  X   X  X      X      X             X
                           \    /           \    /        \           / \
                            X  X             X  X          X         X   X
                                                            \
                                                             X

n이 주어졌을 때, n번째 바이너리 트리를 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있으며, n이 하나 주어진다. (1 ≤ n ≤ 500,000,000) n=0인 경우에 프로그램을 종료하면 된다.

출력

각각의 n에 대해서, 트리를 아래와 같이 출력한다.

  • 자식이 없는 트리는 x로 출력한다.
  • 왼쪽, 오른쪽 서브트리가 L과 R이고, 각각을 출력한 결과가 L', R'일 때, (L')x(R')을 출력한다.
    • L이 비어잇으면 X(R')을 출력한다.
    • R이 비어있으면 (L')x를 출력한다.

예제 입력 1

1
20
31117532
0

예제 출력 1

X
((X)X(X))X
(X(X(((X(X))X(X))X(X))))X(((X((X)X((X)X)))X)X)
[{"problem_id":"7354","problem_lang":"0","title":"\ud2b8\ub9ac\uc758 \uc21c\uc11c","description":"<p>\ub2e4\uc74c\uacfc \uacfc\uc815\uc744 \uac70\uccd0\uc11c \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\uc5d0 \ubc88\ud638\ub97c \ubd99\uc77c \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ube44\uc5b4\uc788\ub294 \ud2b8\ub9ac\uc758 \ubc88\ud638\ub294 0\uc774\ub2e4.<\/li>\r\n\t<li>\ub178\ub4dc\uac00 1\uac1c\uc778 \ud2b8\ub9ac\ub294 1\ubc88\uc774\ub2e4.<\/li>\r\n\t<li>\ub178\ub4dc\uac00 m\uac1c\uc778 \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\uc758 \ubc88\ud638\ub294 m+1\uac1c\uc778 \ud2b8\ub9ac\uc758 \ubc88\ud638\ubcf4\ub2e4 \uc791\ub2e4.<\/li>\r\n\t<li>\ub178\ub4dc\uac00 m\uac1c\uc774\uace0, \uc67c\ucabd\uacfc \uc624\ub978\ucabd \uc11c\ube0c\ud2b8\ub9ac\uac00 \uac01\uac01 L\uacfc R\uc778 \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\uc758 \ubc88\ud638 n\uc740 \uc544\ub798\uc640 \uac19\uc740 \ub450 \uc870\uac74 \uc911 \ud558\ub098\ub77c\ub3c4 \ub9cc\uc871\ud558\ub294 \ud2b8\ub9ac\ubcf4\ub2e4 \ubc88\ud638\uac00 \uc791\ub2e4.&nbsp;\r\n\t<ul>\r\n\t\t<li>\uc67c\ucabd \uc11c\ube0c \ud2b8\ub9ac\uac00 L\ubcf4\ub2e4 \ubc88\ud638\uac00 \ud06c\ub2e4. \ub610\ub294<\/li>\r\n\t\t<li>\uc67c\ucabd \uc11c\ube0c \ud2b8\ub9ac\uac00 L\uacfc \uac19\uace0, \uc624\ub978\ucabd \uc11c\ube0c\ud2b8\ub9ac\uac00 R\ubcf4\ub2e4 \ubc88\ud638\uac00 \ud06c\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>\ucc98\uc74c 10\uac1c \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\uc640 20\ubc88\uc9f8 \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\ub97c \uadf8\ub9bc\uc73c\ub85c \uadf8\ub824\ubcf4\uba74 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<pre>\r\n        0  1  2      3  4      5      6      7      8  9        ...     20\r\n\r\n           X  X      X  X      X      X      X      X  X                 X\r\n               \\    \/    \\      \\    \/ \\    \/      \/    \\               \/\r\n                X  X      X      X  X   X  X      X      X             X\r\n                           \\    \/           \\    \/        \\           \/ \\\r\n                            X  X             X  X          X         X   X\r\n                                                            \\\r\n                                                             X\r\n<\/pre>\r\n\r\n<p>n\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, n\ubc88\uc9f8 \ubc14\uc774\ub108\ub9ac \ud2b8\ub9ac\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc73c\uba70, n\uc774 \ud558\ub098 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; n &le; 500,000,000) n=0\uc778 \uacbd\uc6b0\uc5d0 \ud504\ub85c\uadf8\ub7a8\uc744 \uc885\ub8cc\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 n\uc5d0 \ub300\ud574\uc11c, \ud2b8\ub9ac\ub97c \uc544\ub798\uc640 \uac19\uc774 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc790\uc2dd\uc774 \uc5c6\ub294 \ud2b8\ub9ac\ub294 x\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/li>\r\n\t<li>\uc67c\ucabd, \uc624\ub978\ucabd \uc11c\ube0c\ud2b8\ub9ac\uac00 L\uacfc R\uc774\uace0, \uac01\uac01\uc744 \ucd9c\ub825\ud55c \uacb0\uacfc\uac00 L&#39;, R&#39;\uc77c \ub54c, (L&#39;)x(R&#39;)\uc744 \ucd9c\ub825\ud55c\ub2e4.\r\n\t<ul>\r\n\t\t<li>L\uc774 \ube44\uc5b4\uc787\uc73c\uba74 X(R&#39;)\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/li>\r\n\t\t<li>R\uc774 \ube44\uc5b4\uc788\uc73c\uba74 (L&#39;)x\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"7354","problem_lang":"1","title":"Trees Made to Order","description":"<p>We can number binary trees using the following scheme:<\/p>\r\n\r\n<ul>\r\n\t<li>The empty tree is numbered 0.<\/li>\r\n\t<li>The single-node tree is numbered 1.<\/li>\r\n\t<li>All binary trees having m nodes have numbers less than all those having m + 1 nodes.<\/li>\r\n\t<li>Any binary tree having m nodes with left and right subtrees L and R is numbered n such that all trees having m nodes numbered &gt; n have either\r\n\t<ul>\r\n\t\t<li>left subtrees numbered higher than L, or<\/li>\r\n\t\t<li>a left subtree = L and a right subtree numbered higher than R.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>The first 10 binary trees and tree number 20 in this sequence are shown below:<\/p>\r\n\r\n<pre>\r\n        0  1  2      3  4      5      6      7      8  9        ...     20\r\n\r\n           X  X      X  X      X      X      X      X  X                 X\r\n               \\    \/    \\      \\    \/ \\    \/      \/    \\               \/\r\n                X  X      X      X  X   X  X      X      X             X\r\n                           \\    \/           \\    \/        \\           \/ \\\r\n                            X  X             X  X          X         X   X\r\n                                                            \\\r\n                                                             X\r\n<\/pre>\r\n\r\n<p>Your job for this problem is to output a binary tree when given its order number<\/p>\r\n","input":"<p>Input consists of multiple problem instances. Each instance consists of a single integer n, where 1$ \\le$n$ \\le$500, 000, 000. A value of n = 0 terminates input. (Note that this means you will never have to output the empty tree.)<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<p>For each problem instance, you should output one line containing the tree corresponding to the order number for that instance. To print out the tree, use the following scheme:<\/p>\r\n\r\n<ul>\r\n\t<li>A tree with no children should be output as X.<\/li>\r\n\t<li>A tree with left and right subtrees L and R should be output as (L&#39;)X(R&#39;), where L&#39; and R&#39; are the representations of L and R.\r\n\t<ul>\r\n\t\t<li>If L is empty, just output X(R&#39;).\r\n\t\t<ul>\r\n\t\t\t<li>If R is empty, just output (L&#39;)X.<\/li>\r\n\t\t<\/ul>\r\n\t\t<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]