{"id":4652,"date":"2019-01-29T06:00:07","date_gmt":"2019-01-29T04:00:07","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4652"},"modified":"2019-02-05T08:19:31","modified_gmt":"2019-02-05T06:19:31","slug":"arboles-con-n-elementos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/arboles-con-n-elementos\/","title":{"rendered":"\u00c1rboles con n elementos"},"content":{"rendered":"<p>Los \u00e1rboles binarios se pueden representar con<\/p>\n<pre lang=\"text\"> \n   data Arbol a = H a\n                | N a (Arbol a) (Arbol a)\n     deriving (Show, Eq)\n<\/pre>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\"> \n   arboles  :: Integer -> a -> [Arbol a]\n   nArboles :: [Integer]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(arboles n x) es la lista de todos los \u00e1rboles binarios con n elementos iguales a x. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">   \n     \u03bb> arboles 0 7\n     []\n     \u03bb> arboles 1 7\n     [H 7]\n     \u03bb> arboles 2 7\n     []\n     \u03bb> arboles 3 7\n     [N 7 (H 7) (H 7)]\n     \u03bb> arboles 4 7\n     []\n     \u03bb> arboles 5 7\n     [N 7 (H 7) (N 7 (H 7) (H 7)),N 7 (N 7 (H 7) (H 7)) (H 7)]\n     \u03bb> arboles 6 7\n     []\n     \u03bb> arboles 7 7\n     [N 7 (H 7) (N 7 (H 7) (N 7 (H 7) (H 7))),\n      N 7 (H 7) (N 7 (N 7 (H 7) (H 7)) (H 7)),\n      N 7 (N 7 (H 7) (H 7)) (N 7 (H 7) (H 7)),\n      N 7 (N 7 (H 7) (N 7 (H 7) (H 7))) (H 7),\n      N 7 (N 7 (N 7 (H 7) (H 7)) (H 7)) (H 7)]\n<\/pre>\n<ul>\n<li>nArboles es la sucesi\u00f3n de los n\u00fameros de \u00e1rboles con k elementos iguales a 7, con k \u2208 {1,3,5,&#8230;}. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">   \n     \u03bb> take 14 nArboles\n     [1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900]\n     \u03bb> nArboles !! 100\n     896519947090131496687170070074100632420837521538745909320\n     \u03bb> length (show (nArboles !! 1000))\n     598\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\n\ndata Arbol a = H a\n             | N a (Arbol a) (Arbol a)\n  deriving (Show, Eq)\n\n-- 1\u00aa definici\u00f3n de arboles\n-- ========================\n\narboles :: Integer -> a -> [Arbol a]\narboles 0 _ = []\narboles 1 x = [H x]\narboles n x = [N x i d | k <- [0..n-1],\n                         i <- arboles k x,\n                         d <- arboles (n-1-k) x]\n\n-- 2\u00aa definici\u00f3n de arboles\n-- ========================\n\narboles2 :: Integer -> a -> [Arbol a]\narboles2 0 _ = []\narboles2 1 x = [H x]\narboles2 n x = [N x i d | k <- [1,3..n-1],\n                          i <- arboles2 k x,\n                          d <- arboles2 (n-1-k) x]\n  \n-- 1\u00aa definici\u00f3n de nArboles\n-- =========================\n\nnArboles :: [Integer]\nnArboles = [genericLength (arboles2 n 7) | n <- [1,3..]]\n\n-- 2\u00aa definici\u00f3n de nArboles\n-- =========================\n\n-- Con la definici\u00f3n anterior se observa que nArboles es\n--    1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900\n-- son los n\u00fameros de Catalan https:\/\/en.wikipedia.org\/wiki\/Catalan_number\n-- Una forma de calcularlos (ver https:\/\/oeis.org\/A000108) es\n--     (2n)!\/(n!(n+1)!)\n\nnArboles2 :: [Integer]\nnArboles2 =\n  [factorial (2*n) `div` (factorial n * factorial (n+1)) | n <- [0..]]\n\nfactorial :: Integer -> Integer\nfactorial n = product [1..n]\n\n-- 3\u00aa definici\u00f3n de nArboles\n-- =========================\n\n-- 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900\n\n--  1\n--  1 = 1*1\n--  2 = 1*1  + 1*1\n--  5 = 1*2  + 1*1 + 2*1\n-- 14 = 1*5  + 1*2 + 2*1 + 5*1 \n-- 42 = 1*14 + 1*5 + 2*2 + 5*1 + 14*1\n\nnArboles3 :: [Integer]\nnArboles3 = 1 : aux [1]\n  where aux cs = c : aux (c:cs)\n          where c = sum (zipWith (*) cs (reverse cs))  \n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> length (show (nArboles !! 12))\n--    6\n--    (6.50 secs, 1,060,563,128 bytes)\n--    \u03bb> length (show (nArboles2 !! 12))\n--    6\n--    (0.01 secs, 108,520 bytes)\n--    \u03bb> length (show (nArboles3 !! 12))\n--    6\n--    (0.01 secs, 119,096 bytes)\n--\n--    \u03bb> length (show (nArboles2 !! 1000))\n--    598\n--    (0.01 secs, 4,796,440 bytes)\n--    \u03bb> length (show (nArboles3 !! 1000))\n--    598\n--    (1.66 secs, 321,771,704 bytes)\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nNi vale nada el fruto<br \/>\ncogido sin saz\u00f3n &#8230;<br \/>\nNi aunque te elogie un bruto<br \/>\nha de tener raz\u00f3n.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Los \u00e1rboles binarios se pueden representar con data Arbol a = H a | N a (Arbol a) (Arbol a) deriving (Show, Eq) Definir las funciones arboles :: Integer -> a -> [Arbol a] nArboles :: [Integer] tales que (arboles n x) es la lista de todos los \u00e1rboles binarios con n elementos iguales a&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[7],"tags":[269,8,258,11,6,32,40,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4652"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=4652"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4652\/revisions"}],"predecessor-version":[{"id":4693,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4652\/revisions\/4693"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}