(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 30955, 827]*) (*NotebookOutlinePosition[ 31590, 849]*) (* CellTagsIndexPosition[ 31546, 845]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Computation of the number of binary trees with n nodes", "Title"], Cell[TextData[{ "Here is an example of how a recursive formula is implemented. The \ recursive formula for the number of binary trees with n nodes is\n", StyleBox["Basis", FontWeight->"Bold"], ": b(0)=1" }], "Text"], Cell[TextData[{ "\n", StyleBox["Recursion", FontWeight->"Bold"], ": for ", Cell[BoxData[ \(TraditionalForm\`n > 0\)]], ": ", Cell[BoxData[{ \(TraditionalForm\`b \((n)\) = \(b \((n - 1)\) b \((0)\)\ + \ b \((n - 2)\) \(b(1)\)\ + \ ... \)\ + \ \(b(0)\) \(b( n - 1)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\t\(\(=\)\(\[Sum]\+\(k = 0\)\%\(n - 1\)\(b( n - 1 - k)\) \(b(k)\)\)\)\)}]], "\n" }], "Text"], Cell[TextData[{ "To implement this in ", StyleBox["Mathematica", FontSlant->"Italic"], ", most of the revisions are notational such as ( )'s turn into [ ]'s. \ One crucial thing is the underscore that follows n in the second line:\n" }], "Text"], Cell[TextData[{ "The formula used here for the number of binary trees with n elements can \ be simplified. See \n", ButtonBox["http://mathworld.wolfram.com/BinaryTree.html", ButtonData:>{ URL[ "http://mathworld.wolfram.com/BinaryTree.html"], None}, ButtonStyle->"Hyperlink"], " or \n", ButtonBox["http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/\ eisA.cgi?Anum=A001699", ButtonData:>{ URL[ "http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.\ cgi?Anum=A001699"], None}, ButtonStyle->"Hyperlink"], "\n" }], "Text"], Cell[BoxData[{ \(\(b[0] = 1;\)\), "\n", \(b[n_] := Sum[b[n - 1 - k]\ b[k], {k, 0, n - 1}]\), "\n", \(\)}], "Input"], Cell["\<\ The underscore on the left side of the last expression, turns n \ into a \"Blank\" which allows the formula to be used the way a function is \ intended. You substitute anything into \"Blank\" and the formula on the right \ is calculated with that value. 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1\\)\\) \\(\\(\\(\\(b(\\(\\(\\(\\(-k\\)\\) + 2 \ - 1\\)\\))\\)\\)\\\\ \\(\\(b(k)\\)\\)\\)\\)\\)\\). \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::trace\\\"]\\)\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{\(b::"trace"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\(TraditionalForm\\`\\(b(\\(\\(\\(\\(-k\\)\\) + 2 - 1\\)\\))\\)\\) \ --> \\!\\(TraditionalForm\\`\\(b(1)\\)\\). \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::trace\\\"]\\)\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{\(b::"trace"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\(TraditionalForm\\`\\(b(1)\\)\\) --> \\!\\(TraditionalForm\\`\\(\\(\ \[Sum]\\+\\(k = 0\\)\\%\\(1 - 1\\)\\) \\(\\(\\(\\(b(\\(\\(\\(\\(-k\\)\\) + 1 \ - 1\\)\\))\\)\\)\\\\ \\(\\(b(k)\\)\\)\\)\\)\\)\\). \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::trace\\\"]\\)\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{\(b::"trace"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\(TraditionalForm\\`\\(b(\\(\\(\\(\\(-k\\)\\) + 1 - 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Notice how the times below blow up. The \ following definition saves the computed values makes the computations of \ larger values feasible. Essentially, b[n_] is defined to set (and store) a \ value b[n], where n is what is put into the slot", FontFamily->"Helvetica"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Timing[b[#]] &, Range[0, 12]]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(0.00003999999999981796`\ Second\), "1"}, {\(0.00009500000000017828`\ Second\), "1"}, {\(0.0001389999999998892`\ Second\), "2"}, {\(0.0003969999999999807`\ Second\), "5"}, {\(0.0011610000000001897`\ Second\), "14"}, {\(0.003345999999999849`\ Second\), "42"}, {\(0.010644000000000098`\ Second\), "132"}, {\(0.030187000000000075`\ Second\), "429"}, {\(0.09060200000000007`\ Second\), "1430"}, {\(0.2239`\ Second\), "4862"}, {\(0.49253199999999975`\ Second\), "16796"}, {\(1.4770359999999996`\ Second\), "58786"}, {\(4.430703000000001`\ Second\), "208012"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[ \(Clear[b]\)], "Input"], Cell[BoxData[{ \(\(b[0] = 1;\)\), "\n", \(b[n_] := \(b[n] = Sum[b[n - 1 - k]\ b[k], {k, 0, n - 1}]\)\), "\n", \(\)}], "Input"], Cell["Now the time to compute these values is negligible.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Timing[b[#]] &, Range[0, 12]]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(0.000037999999999094314`\ Second\), "1"}, {\(0.000128000000000128`\ Second\), "1"}, {\(0.00009600000000009601`\ Second\), "2"}, {\(0.0001129999999998077`\ Second\), "5"}, {\(0.0001340000000000785`\ Second\), "14"}, {\(0.00016099999999941161`\ Second\), "42"}, {\(0.00023800000000040455`\ Second\), "132"}, {\(0.00022199999999994446`\ Second\), "429"}, {\(0.00023799999999951638`\ Second\), "1430"}, {\(0.00026500000000062585`\ Second\), "4862"}, {\(0.00029399999999935034`\ Second\), "16796"}, {\(0.0004149999999993881`\ Second\), "58786"}, {\(0.00043700000000068684`\ Second\), "208012"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The logs of these values grow linearly, so that means b is \ exponential in growth.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Log[b /@ Range[25]] // ListPlot\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! 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