시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
2 초 128 MB 65 27 19 33.333%

문제

월드 장난감 회사는 노끈들을 이용해서 트리의 모형을 만들고자 한다. 이를 돕기 위한 프로그램을 작성하라.

트리 (tree)는 그래프 (graph)의 일종으로, 연결되어 있으며, 내부에 싸이클이 존재하지 않는 그래프이다. 그래프에서 버텍스 (vertex), 에지 (edge)라고 부르는 것을, 트리에 한해서는 각각 노드 (node)와 링크 (link)라고 부른다.

트리의 노드들은 노끈 위에 매인 매듭으로 표현되고, 각 매듭 사이를 연결하는 노끈이 트리의 링크들을 표현한다. 노드를 표현하는 매듭들은 그냥 한 개의 노끈 중간이나 끝부분을 묶은 것일 수도 있고, 여러 개의 노끈들이 한 점에서 뭉쳐 묶여 있는 부분일 수도 있다.

기술적인 문제 때문에, 생산 단가는 모델을 만드는 데 사용되는 노끈의 개수와, 사용된 노끈들 중 가장 긴 노끈의 길이에 의존한다. (각각의 링크들은 길이가 모두 1이다. 매듭을 만드는 데 소요되는 노끈의 길이는 무시한다.)

여러분이 할 일은 첫째, 트리의 구조가 주어지면 이를 만들기 위한 최소의 노끈 개수를 구하는 것이다. 둘째로는, 이와 같이 최소 개수의 노끈들을 이용하여 트리를 만들 때 사용되는 가장 긴 노끈 길이의 최솟값을 구하는 것이다.

입력

첫째 줄에 노드의 개수를 나타내는 양의 정수 n이 주어진다. (2≤n≤10,000) 둘째 줄부터는 n-1개의 링크를 나타내는 두 개씩의 양의 정수가 주어진다. 각각은 링크가 연결 짓는 두 노드의 번호를 나타낸다. 노드의 번호는 1번부터 n번까지 빠짐없이 연속하여 붙어 있다.

출력

첫 줄에 필요한 최소한의 노끈 개수와, 가장 긴 노끈 길이의 최솟값을 출력한다.

예제 입력 1

9
7 8
4 5
5 6
1 2
3 2
9 8
2 5
5 8

예제 출력 1

4 2

힌트

[{"problem_id":"1839","problem_lang":"0","title":"\ud2b8\ub9ac \ubaa8\ub378 \ub9cc\ub4e4\uae30","description":"<p>\uc6d4\ub4dc \uc7a5\ub09c\uac10 \ud68c\uc0ac\ub294 \ub178\ub048\ub4e4\uc744 \uc774\uc6a9\ud574\uc11c \ud2b8\ub9ac\uc758 \ubaa8\ud615\uc744 \ub9cc\ub4e4\uace0\uc790 \ud55c\ub2e4. \uc774\ub97c \ub3d5\uae30 \uc704\ud55c \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\ub77c.<\/p><p>\ud2b8\ub9ac (tree)\ub294 \uadf8\ub798\ud504 (graph)\uc758 \uc77c\uc885\uc73c\ub85c, \uc5f0\uacb0\ub418\uc5b4 \uc788\uc73c\uba70, \ub0b4\ubd80\uc5d0 \uc2f8\uc774\ud074\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uadf8\ub798\ud504\uc774\ub2e4. \uadf8\ub798\ud504\uc5d0\uc11c \ubc84\ud14d\uc2a4 (vertex), \uc5d0\uc9c0 (edge)\ub77c\uace0 \ubd80\ub974\ub294 \uac83\uc744, \ud2b8\ub9ac\uc5d0 \ud55c\ud574\uc11c\ub294 \uac01\uac01 \ub178\ub4dc (node)\uc640 \ub9c1\ud06c (link)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p><p>\ud2b8\ub9ac\uc758 \ub178\ub4dc\ub4e4\uc740 \ub178\ub048 \uc704\uc5d0 \ub9e4\uc778 \ub9e4\ub4ed\uc73c\ub85c \ud45c\ud604\ub418\uace0, \uac01 \ub9e4\ub4ed \uc0ac\uc774\ub97c \uc5f0\uacb0\ud558\ub294 \ub178\ub048\uc774 \ud2b8\ub9ac\uc758 \ub9c1\ud06c\ub4e4\uc744 \ud45c\ud604\ud55c\ub2e4. \ub178\ub4dc\ub97c \ud45c\ud604\ud558\ub294 \ub9e4\ub4ed\ub4e4\uc740 \uadf8\ub0e5 \ud55c \uac1c\uc758 \ub178\ub048 \uc911\uac04\uc774\ub098 \ub05d\ubd80\ubd84\uc744 \ubb36\uc740 \uac83\uc77c \uc218\ub3c4 \uc788\uace0, \uc5ec\ub7ec \uac1c\uc758 \ub178\ub048\ub4e4\uc774 \ud55c \uc810\uc5d0\uc11c \ubb49\uccd0 \ubb36\uc5ec \uc788\ub294 \ubd80\ubd84\uc77c \uc218\ub3c4 \uc788\ub2e4.<\/p><p>\uae30\uc220\uc801\uc778 \ubb38\uc81c \ub54c\ubb38\uc5d0, \uc0dd\uc0b0 \ub2e8\uac00\ub294 \ubaa8\ub378\uc744 \ub9cc\ub4dc\ub294 \ub370 \uc0ac\uc6a9\ub418\ub294 \ub178\ub048\uc758 \uac1c\uc218\uc640, \uc0ac\uc6a9\ub41c \ub178\ub048\ub4e4 \uc911 \uac00\uc7a5 \uae34 \ub178\ub048\uc758 \uae38\uc774\uc5d0 \uc758\uc874\ud55c\ub2e4. (\uac01\uac01\uc758 \ub9c1\ud06c\ub4e4\uc740 \uae38\uc774\uac00 \ubaa8\ub450 1\uc774\ub2e4. \ub9e4\ub4ed\uc744 \ub9cc\ub4dc\ub294 \ub370 \uc18c\uc694\ub418\ub294 \ub178\ub048\uc758 \uae38\uc774\ub294 \ubb34\uc2dc\ud55c\ub2e4.)<\/p><p>\uc5ec\ub7ec\ubd84\uc774 \ud560 \uc77c\uc740 \uccab\uc9f8, \ud2b8\ub9ac\uc758 \uad6c\uc870\uac00 \uc8fc\uc5b4\uc9c0\uba74 \uc774\ub97c \ub9cc\ub4e4\uae30 \uc704\ud55c \ucd5c\uc18c\uc758 \ub178\ub048 \uac1c\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc774\ub2e4. \ub458\uc9f8\ub85c\ub294, \uc774\uc640 \uac19\uc774 \ucd5c\uc18c \uac1c\uc218\uc758 \ub178\ub048\ub4e4\uc744 \uc774\uc6a9\ud558\uc5ec \ud2b8\ub9ac\ub97c \ub9cc\ub4e4 \ub54c \uc0ac\uc6a9\ub418\ub294 \uac00\uc7a5 \uae34 \ub178\ub048 \uae38\uc774\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \uac83\uc774\ub2e4.","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ub178\ub4dc\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc591\uc758 \uc815\uc218 n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (2&le;n&le;10,000) \ub458\uc9f8 \uc904\ubd80\ud130\ub294 n-1\uac1c\uc758 \ub9c1\ud06c\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uac1c\uc529\uc758 \uc591\uc758 \uc815\uc218\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01\uac01\uc740 \ub9c1\ud06c\uac00 \uc5f0\uacb0 \uc9d3\ub294 \ub450 \ub178\ub4dc\uc758 \ubc88\ud638\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ub178\ub4dc\uc758 \ubc88\ud638\ub294 1\ubc88\ubd80\ud130 n\ubc88\uae4c\uc9c0 \ube60\uc9d0\uc5c6\uc774 \uc5f0\uc18d\ud558\uc5ec \ubd99\uc5b4 \uc788\ub2e4.","output":"<p>\uccab \uc904\uc5d0 \ud544\uc694\ud55c \ucd5c\uc18c\ud55c\uc758 \ub178\ub048 \uac1c\uc218\uc640, \uac00\uc7a5 \uae34 \ub178\ub048 \uae38\uc774\uc758 \ucd5c\uc19f\uac12\uc744 \ucd9c\ub825\ud55c\ub2e4.","hint":"<p><img width=\"356\" height=\"229\" src=\"\/JudgeOnline\/upload\/201006\/Screen shot 2010-06-11 at 4_52_18 PM.png\" alt=\"\" \/><\/p>","original":"0","html_title":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1839","problem_lang":"1","title":"Strings","description":"<p>String-Toys joint-stock company has asked you for help in solving the following problem. They want to manufacture models of connected acyclic graphs made of string.<\/p>\r\n\r\n<p>Each graph consists of vertices and a certain number of edges joining distinct vertices. Each vertex can be joined to many other vertices. A graph is connected and acyclic, if every vertex can be reached from every other vertex by travelling along the edges and furthermore there is a unique way of doing so without turning back.<\/p>\r\n\r\n<p>Graphs&#39; vertices are to be modelled by nodes knotted on bits of string, while edges by the bits of string themselves between the nodes. Each node can be either a knot on a single bit of string or many bits knotted at one point. Due to technical reasons cost of production depends on the number of string bits used in making a model and length of the longest bit. (Each edge has length of 1. Length of string used to making a knot is negligible.)<\/p>\r\n\r\n<p>You are to write a programme that:<\/p>\r\n\r\n<ul>\r\n\t<li>reads from the standard input the description of a connected acyclic graph to be modelled,<\/li>\r\n\t<li>calculates:\r\n\t<ul>\r\n\t\t<li>the minimal number of string bits necessary to making the model,<\/li>\r\n\t\t<li>assuming that the number of string bit is minimal, the minimal length of the longest string bit needed to making the model.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>writes the obtained numbers to the standard output.<\/li>\r\n<\/ul>\r\n","input":"<p>The first line contains one positive integer n - the number of vertices in the graph, 2 &le; n &le; 10 000. Vertices are numbered from 1 to n. The following n-1 lines contain descriptions of edges, one per line. Each of these lines contains a pair of integers a and b separated by a single space, 1 &le; a,b &le; n, a &ne; b. Such pair represents an edge joining vertices a and b.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<p>Your programme should write two integers k and l separated by a single space in the first line of the output: k should be the minimal number of string bits required to make the graph&#39;s model, while l should be the minimal length of the longest string bit (assuming that k string bits are used).<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","hint":"<p><img alt=\"\" src=\"\/upload\/images\/szn.gif\" style=\"height:190px; width:302px\" \/><\/p>\r\n","original":"1","html_title":"0","problem_lang_code":"\uc601\uc5b4"}]