Manipulating sets of hierarchical data

• US 20050187900A1
• Filed: 12/06/2004
• Published: 08/25/2005
• Est. Priority Date: 02/09/2004
• Status: Active Grant

• Current Assignee: Lower48 IP, LLC (Lower48 Analytics, Inc.), Virginia T. Jenkins
• Original Assignee: Robert T. and Virginia T. Jenkins as Trustees for the Jenkins Family Trust Dated February 8, 2002
• Inventors: LeTourneau, Jack J.
• Granted Patent: US 8,037,102 B2
• Time in Patent Office: Days
• US Class Current: 1/1
• CPC Class Codes: G06F 16/2246 Trees, e.g. B+trees
• CPC Class Codes: G06F 16/258 Data format conversion from or to a database
• CPC Class Codes: G06F 16/282 Hierarchical databases, e.g. IMS, LDAP data stores or Lotus Notes

Abstract

Embodiments of methods, apparatuses, devices and/or systems for manipulating hierarchical sets of data are disclosed.

Claims

10. A method of combining at least two trees comprising:

converting said at least two trees to strings;

combining said strings; and

converting the combination of strings to a tree.

11. The method of claim 10, wherein said trees comprise binary edge labeled trees.

12. The method of claim 10, wherein said at least two trees are converted to binary edge labeled trees.

13. The method of claim 10, wherein at least one of said at least two trees includes at least one null label.

14. The method of claim 13, wherein said at least one of said at least two trees that includes at least one null label is converted to a tree without a null label.

15. The method of claim 14, wherein said tree without a null label is converted to a binary edge labeled tree.

16. The method of claim 10, wherein said at least two trees are converted to binary node labeled trees.

17. The method of claim 10, wherein said strings comprise binary strings.

18. The method of claim 17, wherein said binary strings comprise binary edge labeled strings.

19. The method of claim 10, wherein said strings comprise binary strings, and wherein combining said binary strings comprises:

converting said binary strings to binary numerals, multiplying said binary numerals, and converting the product to a binary string.

20. A method of splitting a first tree into at least second and third trees comprising:

converting said first tree to a first numeral;

factoring said first numeral into at least second and third numerals; and

converting said at least second and third numerals to at least second and third trees.

21. The method of claim 20, wherein said first tree comprises a binary edge labeled tree.

22. The method of claim 20, wherein said first tree is converted to a binary edge labeled tree.

23. The method of claim 20, wherein said first tree includes at least one null label.

24. The method of claim 23, wherein said first tree is converted to a tree without a null label.

25. The method of claim 24, wherein said tree without a null label is converted to a binary edge labeled tree.

26. The method of claim 20, wherein said first tree is converted to a binary node labeled tree.

27. The method of claim 20, wherein said numerals comprise binary numerals.

28. The method of claim 20, wherein said numerals are converted to binary numerals.

29. A method of splitting a first tree into at least second and third trees comprising:

converting said first tree to a first string;

splitting said first string into at least second and third strings; and

converting said at least second and third strings to said at least second and third trees.

30. The method of claim 29, wherein said first tree comprises a binary edge labeled tree.

31. The method of claim 29, wherein said first tree is converted to a binary edge labeled tree.

32. The method of claim 29, wherein said first tree includes at least one null label.

33. The method of claim 32, wherein said first tree is converted to a tree without a null label.

34. The method of claim 33, wherein said tree without a null label is converted to a binary edge labeled tree.

35. The method of claim 32, wherein said tree without a null label is converted to a binary node labeled tree.

36. The method of claim 29, wherein said first tree is converted to a binary node labeled tree.

37. The method of claim 29, wherein said strings comprise binary strings.

38. The method of claim 37, wherein said binary strings are converted to numerals.

39. The method of claim 37, wherein said binary strings comprise binary edge labeled strings.

40. The method of claim 29, wherein said strings comprise binary strings, and wherein splitting said first string into at least second and third strings comprises:

converting said first strings to binary numerals, factoring said binary numerals, and converting the factors to binary strings.

41. A method of converting between a string and a tree comprising:

converting said string to a numeral;

converting said numeral to said tree.

42. The method of claim 41, wherein said string comprises a binary string.

43. The method of claim 42, wherein said binary string comprises a binary edge labeled string.

44. The method of claim 41, wherein said tree comprises a binary edge labeled tree.

45. The method of claim 41, wherein said tree comprises a binary node labeled tree.

46. The method of claim 45, wherein said binary node labeled tree is converted to a numeral node labeled tree.

47. The method of claim 46, wherein said numeral node labeled tree is converted to a numeral node labeled tree with at least one null label.

48. The method of claim 41, wherein said string is converted to a binary edge labeled string.

49. The method of claim 41, wherein said tree is converted to a tree with at least one null label.

50. The method of claim 41, wherein said tree comprises a binary edge labeled tree, wherein said binary edge labeled tree is converted to a numeral edge labeled tree.

51. The method of claim 50, wherein said binary edge labeled tree is converted to a numeral edge labeled tree with at least one null label.

52. The method of claim 41, wherein said numeral comprises a binary numeral.

53. The method of claim 41, wherein said numeral is converted to a binary numeral.

54. A method of converting between a tree and a string comprising:

converting said tree to a numeral;

converting said numeral to said string.

55. The method of claim 54, wherein said string comprises a binary string.

56. The method of claim 55, wherein said binary string comprises a binary edge labeled string.

57. The method of claim 54, wherein said tree comprises a binary edge labeled tree.

58. The method of claim 54, wherein said tree comprises a binary node labeled tree.

59. The method of claim 54, wherein said tree comprises a numeral node labeled tree, said numeral node labeled tree being converted to a binary node labeled tree.

60. The method of claim 54, wherein said tree comprises a numeral node labeled tree with at least one null label, said numeral labeled tree being converted to a numeral node labeled tree without a null label.

61. The method of claim 54, wherein said tree comprises a numeral edge labeled tree, wherein said numeral edge labeled tree is converted to a binary edge labeled tree.

62. The method of claim 54, wherein said numeral comprises a binary numeral.

63. A method of ordering a set of binary strings comprising:

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based at least upon their associated binary values when a binary one is added to the left of the binary string.

64. The method of claim 63, wherein said binary strings comprise binary node labeled strings;

and

wherein ordering binary node labeled strings of the same length comprises ordering binary node labeled strings based at least upon the associated binary values.

65. The method of claim 63, wherein said binary strings comprise binary edge labeled strings;

and

wherein ordering binary edge labeled strings of the same length comprises ordering binary edge labeled strings based at least upon the associated binary values.

66. A method of enumerating a set of trees comprising:

positioning a tree at location k, where k is the product of u and v, such that said tree is formed by a union of trees in positions u and v;

k, u, and v comprising positive natural numerals greater than one.

67. The method of claim 66, wherein said trees comprise binary edge labeled trees.

68. The method of claim 66, wherein said trees comprise binary node labeled trees.

69. The method of claim 66, wherein said trees comprise numeral edge labeled trees,

and further comprising:

converting said numeral edge labeled trees to binary edge labeled trees.

70. The method of claim 66, wherein said trees comprise numeral node labeled trees,

and further comprising:

converting said numeral node labeled trees to binary node labeled trees.

71. The method of claim 66, and further comprising:

positioning a tree at location j, where j comprises a non-composite numeral greater than one, such that said tree is formed from a prior tree by attaching a node as a root node.

72. The method of claim 71, wherein attaching a node as a root node further comprises labeling the edge between said node and said root node.

73. The method of claim 72, wherein said trees comprise binary edge labeled trees, and said edge is labeled either 0 or 1.

74. The method of claim 73, wherein said edge is labeled with a 0, j being P(2m−

1), where P(2m−

1) is the (2m−

1)th non-composite numeral of the Keene enumeration of non-composite numerals on positive natural numerals.

75. The method of claim 73, wherein said edge is labeled with a 1, j being P(2m), where P(2m) is the (2m)th non-composite numeral of the Keene enumeration of non-composite numerals on positive natural numerals.

76. The method of claim 66, wherein said trees comprise numeral edge labeled trees, and further comprising:

converting said numeral edge labeled trees to binary edge labeled trees.

77. The method of claim 66, and further comprising:

positioning at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted.

78. The method of claim 77, wherein said string comprises a binary string.

79. The method of claim 78, wherein said binary string comprises a binary edge labeled string.

80. The method of claim 78, wherein said binary strings comprises a binary node labeled strings.

81. A method of forming trees comprising:

forming at least one of a zero-push and a one-push of a tree.

82. The method of claim 81, wherein said tree comprises binary edge labeled tree.

83. The method of claim 81, wherein forming trees also comprises forming at least an N-push of said tree, where N is a positive natural numeral greater than one.

84. The method of claim 81, wherein said tree is converted to a binary edge labeled tree from a binary node labeled tree.

85. The method of claim 81, wherein said tree comprises a numeral edge labeled tree.

86. The method of claim 85, wherein said tree is converted to a binary edge labeled tree from said numeral edge labeled tree.

87. The method of claim 85, wherein said numeral edge labeled tree includes at least one null label.

88. The method of claim 81, wherein said tree comprises a numeral node labeled tree.

89. The method of claim 88, wherein said tree is converted to a binary edge labeled tree from said numeral node labeled tree.

90. The method of claim 88, wherein said numerical node labeled tree includes at least one null label.

91. The method of claim 81, and further comprising:

positioning said zero-push tree at position P(2k−

1), if k is the position of said tree in a tree enumeration, P(2k−

1) comprising the (2k−

1)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

92. The method of claim 81, and further comprising:

positioning said one-push tree at position P(2k), if k is the position of said tree in a tree enumeration, P(2k) comprising the (2k)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

93. A method of representing a graphical hierarchy of data using a one-dimensional arrangement of symbols comprising:

employing a first symbol to represent a node or an edge;

employing additional symbols to represent labels of said node or edge;

wherein the successive order of said symbols provides said graphical hierarchy.

94. The method of claim 93, wherein said first symbol represents a node;

said additional symbols comprising two symbols that respectively represent labels 0 and 1; and

wherein said graphical hierarchy comprises a binary edge labeled tree.

95. The method of claim 94, wherein an edge label symbol is provided prior to the symbol for a node associated with the edge that it labels.

96. The method of claim 95, wherein an edge label symbol is provided immediately prior to the symbol for a node associated with the edge that it labels

97. The method of claim 93, wherein said first symbol represents an edge;

said additional symbols comprising two symbols to respectively represent 0 and 1; and

wherein said graphical hierarchy comprises a binary node labeled tree.

98. The method of claim 97, wherein a node label symbol is provided prior to the symbol for an edge associated with the node that it labels.

99. The method of claim 98, wherein a node label symbol is provided immediately prior to the symbol for an edge associated with the node that it labels.

100. The method of claim 93, wherein adjacent label symbols represent successively connected nodes or edges.

101. The method of claim 93, wherein non-adjacent label symbols represent nodes or edges that are not successively connected.

102. An article comprising:

a storage medium having stored thereon an association between strings and tree;

said association comprising a set of trees and strings, respectively having been enumerated by;

positioning a tree at location k, where k is the product of u and v, such that said tree is formed by a union of trees in positions u and v;

k, u, and v comprising positive natural numerals greater than one.

103. The article of claim 102, wherein said trees comprise binary edge labeled trees.

104. The article of claim 102, wherein said trees comprise binary node labeled trees.

105. The article of claim 102, wherein said trees comprise numeral edge labeled trees, said numeral edge labeled trees having been converted to binary edge labeled trees.

106. The article of claim 102, wherein said trees comprise numeral node labeled trees, said numeral node labeled trees having been converted to binary node labeled trees.

107. The article of claim 102, wherein said association having been enumerated by:

positioning a tree at location k, where k comprises a non-composite natural numeral greater than one, such that said tree is formed from a prior tree by attaching a node as a root node.

108. The article of claim 102, wherein said association having been enumerated by:

labeling the edge between said node and said root node.

109. The article of claim 108, wherein said trees comprise binary edge labeled trees, and wherein said association having been enumerated by:

labeling said edge either 0 or 1.

110. The article of claim 109, wherein said association having been enumerated by:

said edge being labeled with a 0.

111. The article of claim 109, wherein said association having been enumerated by:

said edge being labeled with a 1.

112. The article of claim 102, wherein said association having been enumerated by:

positioning at location k a string such that said string corresponding to a binary value, said binary value corresponds to k with the left most binary digit omitted.

113. The article of claim 112, wherein said string comprises a binary string.

114. The article of claim 113, wherein said binary string comprises a binary edge labeled string.

115. The article of claim 113, wherein said binary string comprises a binary node labeled string.

116. A method of enumerating a set of strings comprising:

positioning at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted;

k comprising a positive natural number greater than one.

117. The method of claim 116, wherein said strings comprise binary strings.

118. The method of claim 117, wherein said binary strings comprise binary edge labeled strings.

119. The method of claim 117, wherein said binary strings comprise binary node labeled strings.

120. A method of ordering a set of trees comprising:

converting said set of trees to numerals;

ordering said numerals; and

converting said numerals to said set of trees.

121. The method of claim 120, wherein said trees comprise binary edge labeled trees.

122. The method of claim 120, wherein said trees comprise binary node labeled trees.

123. The method of claim 120, wherein said trees comprise numeral edge labeled trees,

and further comprising:

converting said numeral edge labeled trees to binary edge labeled trees.

124. The method of claim 120, wherein said trees comprise numeral node labeled trees,

and further comprising:

converting said numeral node labeled trees to binary node labeled trees.

125. The method of claim 120, and further comprising ordering a set of strings comprising:

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based upon their associated binary values when a binary one is added to the left of the binary string;

wherein a relationship exists between the ordered trees and the ordered binary strings

126. The method of claim 125, wherein said binary strings comprise binary node labeled strings;

and

wherein ordering binary node labeled strings of the same length comprises ordering binary node labeled strings based at least in part upon the associated binary values.

127. The method of claim 125, wherein said binary strings comprise binary edge labeled strings;

and

wherein ordering binary edge labeled strings of the same length comprises ordering binary edge labeled strings based at least in part upon the associated binary values.

128. An article comprising:

a storage medium having stored thereon an association between a set of strings and a set of trees;

said set of trees having been ordered by;

converting said set of trees to numerals;

ordering said numerals; and

converting said numerals to said set of trees.

129. The article of claim 128, wherein said trees comprise binary edge labeled trees.

130. The article of claim 128, wherein said trees comprise binary node labeled trees.

131. The article of claim 128, wherein said trees comprise numeral edge labeled trees,

said numeral edge labeled trees having been converted to binary edge labeled trees.

132. The article of claim 128, wherein said trees comprise numeral node labeled trees, said numeral node labeled trees having been converted to binary node labeled trees.

133. The article of claim 128, and further comprising ordering a set of strings comprising:

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based upon their associated binary values when a binary one is added to the left of the binary string;

wherein a relationship exists between the ordered trees and the ordered binary strings.

134. The article of claim 133, wherein said binary strings comprise binary node labeled strings;

and

wherein ordering binary node labeled strings of the same length comprises ordering binary node labeled strings based upon the associated binary values.

135. The article of claim 133, wherein said binary strings comprise binary edge labeled strings;

and

wherein ordering binary edge labeled strings of the same length comprises ordering binary edge labeled strings based upon the associated binary values.

136. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert at least two trees to numerals, multiply said numerals, and convert the product of said numerals to a tree.

137. The apparatus of claim 136, wherein said trees comprise binary edge labeled trees.

138. The apparatus of claim 136, wherein said computing platform is adapted to convert said at least two trees to binary edge labeled trees.

139. The apparatus of claim 136, wherein at least one of said at least two trees includes at least one null label.

140. The apparatus of claim 139, wherein said computing platform is adapted to convert said at least one of said at least two trees that includes at least one null label to a tree without a null label.

141. The apparatus of claim 140, wherein said computing platform is adapted to convert said tree without a null type to a binary edge labeled tree.

142. The apparatus of claim 136, wherein said computing platform is adapted to convert said at least two trees to binary node labeled trees.

143. The apparatus of claim 136, wherein said numerals comprise binary numerals.

144. The apparatus of claim 136, wherein said computing platform is adapted to convert said numerals to binary numerals.

145. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert at least two trees to strings, combine said strings, and convert the combination of strings to a tree.

146. The apparatus of claim 145, wherein said trees comprise binary edge labeled trees.

147. The apparatus of claim 145, wherein said computing platform is adapted to convert said at least two trees to binary edge labeled trees.

148. The apparatus of claim 145, wherein at least one of said at least two trees includes at least one null label.

149. The apparatus of claim 148, wherein said computing platform is adapted to convert said at least one of said at least two trees that includes at least one null label to a tree without a null label.

150. The apparatus of claim 149, wherein said computing platform is adapted to convert said tree without a null label to a binary edge labeled tree.

151. The apparatus of claim 145, wherein said computing platform is adapted to convert said at least two trees to binary node labeled trees.

152. The apparatus of claim 145, wherein said strings comprise binary strings.

153. The apparatus of claim 152, wherein said binary strings comprise binary edge labeled strings.

154. The apparatus of claim 145, wherein said strings comprise binary strings, and wherein said computing platform is adapted to combine said binary strings by converting said binary strings to binary numerals, multiplying said binary numerals, and converting the product to a binary string.

155. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert a first tree to a first numeral, factor said first numeral into at least second and third numerals, and convert said second and third numerals to second and third trees.

156. The apparatus of claim 155, wherein said first tree comprises a binary edge labeled tree.

157. The apparatus of claim 155, wherein said computing platform is adapted to convert said first tree to a binary edge labeled tree.

158. The apparatus of claim 155, wherein said first tree includes at least one null label.

159. The apparatus of claim 158, wherein said computing platform is adapted to convert said first tree to a tree without a null label.

160. The apparatus of claim 155, wherein said computing platform is adapted to convert said first tree to a binary node labeled tree.

161. The apparatus of claim 155, wherein said numerals comprise binary numerals.

162. The apparatus of claim 155, wherein said computing platform comprises a computer.

163. The apparatus of claim 155, wherein said computing platform is adapted to convert said numerals to binary numerals.

164. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert a first tree to a first string, split said first string into at least second and third strings, and convert said second and third strings to second and third trees.

165. The apparatus of claim 164, wherein said first tree comprises a binary edge labeled tree.

166. The apparatus of claim 165, wherein said computing platform is adapted to convert said first tree to a binary edge labeled tree.

167. The apparatus of claim 164, wherein said first tree includes at least one null label.

168. The apparatus of claim 166, wherein said computing platform is adapted to convert said first tree to a tree without a null label.

169. The apparatus of claim 168, wherein said computing platform is adapted to convert said tree without a null label to a binary edge labeled tree.

170. The apparatus of claim 167, wherein said computing platform is adapted to convert said tree without a null label to a binary node labeled tree.

171. The apparatus of claim 164, wherein said computing platform is adapted to convert said first tree to a binary node labeled tree.

172. The apparatus of claim 164, wherein said strings comprise binary strings.

173. The apparatus of claim 172, wherein said computing platform is adapted to convert said binary strings to numerals.

174. The apparatus of claim 172, wherein said binary strings comprise binary edge labeled strings.

175. The apparatus of claim 164, wherein said strings comprise binary strings, and wherein said computing platform is adapted to split said first string to at least second and third strings by:

converting said first string to a binary numeral, factoring said binary numeral, and converting the factors to binary strings.

176. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert a string to a numeral, and convert said numeral to a tree.

177. The apparatus of claim 176, wherein said string comprises a binary string.

178. The apparatus of claim 177, wherein said binary string comprises a binary edge labeled string.

179. The apparatus of claim 176, wherein said tree comprises a binary edge labeled tree.

180. The apparatus of claim 176, wherein said tree comprises a binary node labeled tree.

181. The apparatus of claim 180, wherein said computing platform is adapted to convert said binary node labeled to a numeral node labeled tree.

182. The apparatus of claim 181, wherein said computing platform is adapted to convert said numerical node labeled tree to a numeral node labeled tree with at least one null label.

183. The apparatus of claim 176, wherein said computing platform is adapted to convert said string to a binary edge labeled string.

184. The apparatus of claim 176, wherein said computing platform is adapted to convert said tree to a tree with at least one null label.

185. The apparatus of claim 176, wherein said tree comprises a binary edge labeled tree, wherein said computing platform is adapted to convert said binary edge labeled tree to a numeral edge labeled tree.

186. The apparatus of claim 185, wherein said computing platform is adapted to convert said binary edge labeled tree to a numeral edge labeled tree with at least one null label.

187. The apparatus of claim 176, wherein said numeral comprises a binary numeral.

188. The apparatus of claim 176, wherein said computing platform is adapted to convert said numeral to a binary numeral.

189. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert a tree to a numeral, and convert said numeral to a string.

190. The apparatus of claim 189, wherein said string comprises a binary string.

191. The apparatus of claim 190, wherein said binary string comprises a binary edge labeled string.

192. The apparatus of claim 189, wherein said tree comprises a binary edge labeled tree.

193. The apparatus of claim 189, wherein said tree comprises a binary node labeled tree.

194. The apparatus of claim 189, wherein said tree comprises a numeral node labeled tree, wherein said computing platform is adapted to convert said numeral node labeled tree to a binary node labeled tree.

195. The apparatus of claim 189, wherein said tree comprises a numeral node labeled tree with at least one null label, wherein said computing platform is adapted to convert said numeral labeled tree to a numeral node labeled tree without a null label.

196. The apparatus of claim 189, wherein said tree comprises a numeral edge labeled tree, wherein said computing platform is adapted to convert said numeral edge labeled tree to a binary edge labeled tree.

197. The apparatus of claim 189, wherein said numeral comprises a binary numeral.

198. An apparatus comprising:

a computing platform;

said computing platform being adapted to order binary strings by;

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based at least upon their associated binary values when a binary one is added to the left of the binary string.

199. The apparatus of claim 198, wherein said binary strings comprise binary edge labeled strings;

and

wherein said computing platform is adapted to order binary edge labeled strings of the same length based at least upon the associated binary values.

200. The apparatus of claim 198, wherein said binary strings comprise binary node labeled strings;

and

wherein said computing platform is adapted to order binary node labeled strings of the same length based at least upon the associated binary values.

201. An apparatus comprising:

a computing platform;

said platform being adapted to enumerate trees by;

positioning a tree at location k, where k is the product of u and v, such that said tree is formed by a union of trees in positions u and v;

k, u, and v comprising positive natural numerals greater than one.

202. The apparatus of claim 201, wherein said trees comprise binary edge labeled trees.

203. The apparatus of claim 201, wherein said trees comprise binary node labeled trees.

204. The apparatus of claim 201, wherein said trees comprise numeral edge labeled trees,

and wherein said computing platform is adapted to convert said numeral edge labeled trees to binary edge labeled trees.

205. The apparatus of claim 201, wherein said trees comprise numeral node labeled trees,

and wherein said computing platform is adapted to convert said numeral node labeled trees to binary node labeled trees.

206. The apparatus of claim 201, wherein said computing platform is adapted to position a tree at location j, where j comprises a non-composite numeral greater than one, such that said tree is formed from a prior tree by attaching a node as a root node.

207. The apparatus of claim 206, wherein computing platform being adapted to label the edge between said node and said root node.

208. The apparatus of claim 207, wherein said trees comprise binary edge labeled trees, and wherein said computing platform is adapted to label said edge either 0 or 1.

209. The apparatus of claim 208, wherein said computing platform is adapted to label said edge with a 0, j being P(2m−

1), where P(2m−

1) is the (2m−

1)th non-composite numeral of the Keene enumeration of non-composite numerals on positive natural numerals.

210. The apparatus of claim 208, wherein said computing platform is adapted to label said edge with a 1, j being P(2m), where P(2m) is the (2m)th non-composite numeral of the Keene enumeration of non-composite numerals on positive natural numerals.

211. The apparatus of claim 201, wherein said trees comprise numeral edge labeled trees, and wherein said computing platform is adapted to convert said numeral edge labeled trees to binary edge labeled trees.

212. The apparatus of claim 201, wherein said computing platform is adapted to position at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted.

213. The apparatus of claim 212, wherein said strings comprise binary strings.

214. The apparatus of claim 213, wherein said binary strings comprise binary edge labeled strings.

215. The apparatus of claim 213, wherein said binary strings comprise binary node labeled strings:

216. An apparatus comprising:

a computing platform;

said computing platform being adapted to form at least one of a zero-push and a one-push of a tree.

217. The apparatus of claim 216, wherein said tree comprises a binary edge labeled tree.

218. The apparatus of claim 216, wherein said computing platform is adapted to form a tree comprising at least an N-push of said tree, where N is a positive natural numeral greater than one.

219. The apparatus of claim 216, wherein said computing platform is adapted to convert said tree to a binary edge labeled tree.

220. The apparatus of claim 219, wherein said computing platform is adapted to convert said tree to said binary edge labeled tree from a binary node labeled tree.

221. The apparatus of claim 216, wherein said tree comprises a numeral edge labeled tree.

222. The apparatus of claim 221, wherein said computing platform is adapted to convert said tree to a binary edge labeled tree from said numeral edge labeled tree.

223. The apparatus of claim 221, wherein said numeral edge labeled tree includes at least one null label.

224. The apparatus of claim 216, wherein said tree comprises a numeral node labeled tree.

225. The apparatus of claim 224, wherein said computing platform is adapted to convert said tree to a binary edge labeled tree from said numeral node labeled tree.

226. The apparatus of claim 224, wherein said numerical node labeled tree includes at least one null label.

227. An apparatus comprising:

a computing platform;

said computing platform being adapted to employ a first symbol to represent a node or an edge, and employ additional symbols to represent labels of said node or edge, wherein the successive order of said symbols provides a graphical hierarchy.

228. The apparatus of claim 227, wherein said first symbol represents a node;

said additional symbols comprise two symbols that respectively represent labels 0 and 1; and

wherein said graphical hierarchy comprises a binary edge labeled tree.

229. The apparatus of claim 228, wherein said computing platform is adapted to provide an edge label symbol prior to the symbol for a node associated with the edge that it labels.

230. The apparatus of claim 229, wherein said computing platform is adapted to provide an edge label symbol immediately prior to the symbol for a node associated with the edge that it labels

231. The apparatus of claim 227, wherein said first symbol represents an edge;

said additional symbols comprise two symbols to respectively represent 0 and 1; and

wherein said graphical hierarchy comprises a binary node labeled tree.

232. The apparatus of claim 231, wherein said computing platform is adapted to provide a node label symbol prior to the symbol for an edge associated with the node that it labels.

233. The apparatus of claim 232, wherein said computing platform is adapted to provide a node label symbol immediately prior to the symbol for an edge associated with the node that it labels.

234. The apparatus of claim 227, wherein said computing platform is adapted to provide adjacent label symbols to represent successively connected nodes or edges.

235. The apparatus of claim 227, wherein said computing platform is adapted to provide non-adjacent label symbols to represent nodes or edges that are not successively connected.

236. An apparatus comprising:

a computing platform;

said computing platform being adapted to position at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted.

237. The apparatus of claim 236, wherein said strings comprise binary strings.

238. The apparatus of claim 237, wherein said binary strings comprise binary edge labeled strings.

239. The apparatus of claim 237, wherein said binary strings comprise binary node labeled string.

240. An apparatus comprising:

a computing platform;

said computing platform being adapted to convert a set of trees to numerals, order said numerals, and convert said numerals to said set of trees.

241. The apparatus of claim 240, wherein said trees comprise binary edge labeled trees.

242. The apparatus of claim 240, wherein said trees comprise binary node labeled trees.

243. The apparatus of claim 240, wherein said trees comprise numeral edge labeled trees,

and wherein said computing platform is adapted to convert said numeral edge labeled trees to binary edge labeled trees.

244. The apparatus of claim 240, wherein said trees comprise numeral node labeled trees,

and wherein said computing platform is adapted to convert said numeral node labeled trees to binary node labeled trees.

245. The apparatus of claim 240, wherein said computing platform is adapted to order a set of binary strings by:

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based upon their associated binary values when a binary one is added to the left of the binary string;

wherein a relationship exists between the ordered trees and the ordered binary strings.

246. The apparatus of claim 245, wherein said binary strings comprise binary node labeled strings;

and

wherein said computing platform is adapted to order binary node labeled strings of the same length based at least in part upon the associated binary values.

247. The apparatus of claim 245, wherein said binary strings comprise binary edge labeled strings;

and

wherein said computing platform is adapted to order binary edge labeled strings of the same length based at least in part upon the associated binary values.

248. A set of hierarchical data comprising:

a tree having been formed by;

converting at least two other trees to numerals;

multiplying said numerals; and

converting the product of said numerals to said tree.

249. The set of hierarchical data of claim 248, wherein said trees comprise binary edge labeled trees.

250. The set of hierarchical data of claim 248, wherein said tree having been formed by:

said at least two other trees being converted to binary edge labeled trees.

251. The set of hierarchical data of claim 248, wherein tree having been formed by:

at least one of said at least two other trees including at least one null label.

252. The set of hierarchical data of claim 251, wherein said tree having been formed by:

said at least one of said at least two other trees that includes at least one null label being converted to a tree without a null label.

253. The set of hierarchical data of claim 252, wherein said tree having been formed by:

said tree without a null type being converted to a binary edge labeled tree.

254. The set of hierarchical data of claim 248, wherein said tree having been formed by:

said at least two other trees being converted to binary node labeled trees.

255. The set of hierarchical data of claim 248, wherein said numerals comprise binary numerals.

256. The set of hierarchical data of claim 248, wherein said numerals are converted to binary numerals.

257. A set of hierarchical data comprising:

a tree having been formed by;

converting at least two other trees to strings;

combining said strings; and

converting the combination of strings to said tree.

258. The set of hierarchical data of claim 257, wherein said trees comprise binary edge labeled trees.

259. The set of hierarchical data of claim 257, wherein said tree having been formed by:

said at least two other trees being converted to binary edge labeled trees.

260. The set of hierarchical data of claim 257, wherein said tree having been formed by:

at least one of said at least two other trees including at least one null label.

261. The set of hierarchical data of claim 260, wherein said tree having been formed by:

said at least one of said at least two other trees that includes at least one null label being converted to a tree without a null label.

262. The set of hierarchical data of claim 261, wherein said tree having been formed by:

said tree without a null label being converted to a binary edge labeled tree.

263. The set of hierarchical data of claim 257, wherein said tree having been formed by:

said at least two other trees being converted to binary node labeled trees.

264. The set of hierarchical data of claim 257, wherein said strings comprise binary strings.

265. The set of hierarchical data of claim 264, wherein said binary strings comprise binary edge labeled strings.

266. The set of hierarchical data of claim 257, wherein said strings comprise binary strings, and wherein said combined binary strings are formed by:

converting said binary strings to binary numerals, multiplying said binary numerals, and converting the product to a binary string.

267. A set of hierarchical data comprising:

a first and second tree having been formed by;

converting a third tree to a first numeral;

factoring said first numeral into at least second and third numerals; and

converting said second and third numerals to said first and second trees.

268. The set of hierarchical data of claim 267, wherein said third tree comprises a binary edge labeled tree.

269. The set of hierarchical data of claim 267, wherein said first and second trees having been formed by:

said third tree being converted to a binary edge labeled tree.

270. The set of hierarchical data of claim 267, wherein said first and second trees having been formed by:

said third tree including at least one null label.

271. The set of hierarchical data of claim 270, wherein said first and second trees having been formed by:

said third tree being converted to a tree without a null label.

272. The set of hierarchical data of claim 271, wherein said first and second trees having been formed by:

said tree without a null label being converted to a binary edge labeled tree.

273. The set of hierarchical data of claim 267, wherein said first and second trees having been formed by:

said third tree being converted to a binary node labeled tree.

274. The set of hierarchical data of claim 267, wherein said numerals comprise binary numerals.

275. The set of hierarchical data of claim 267, wherein said first and second trees having been formed by:

said numerals being converted to binary numerals.

276. A set of hierarchical data comprising:

a first and second tree having been formed by;

converting a third tree to a first string;

splitting said first string into at least second and third strings;

converting said second and third strings to said first and second trees.

277. The set of hierarchical data of claim 276, wherein said third tree comprises a binary edge labeled tree.

278. The set of hierarchical data of claim 276, wherein said first and second trees having been formed by:

said third tree being converted to a binary edge labeled tree.

279. The set of hierarchical data of claim 276, wherein s said first and second trees having been formed by:

said third tree including at least one null label.

280. The set of hierarchical data of claim 279, wherein said first and second trees having been formed by:

said third tree being converted to a tree without a null label.

281. The set of hierarchical data of claim 280, wherein said first and second trees having been formed by:

said tree without a null label being converted to a binary edge labeled tree.

282. The set of hierarchical data of claim 280, wherein said first and second trees having been formed by:

said tree without a null label being converted to a binary node labeled tree.

283. The set of hierarchical data of claim 276, wherein said first and second trees having been formed by:

said third tree being converted to a binary node labeled tree.

284. The set of hierarchical data of claim 276, wherein said strings comprise binary strings.

285. The set of hierarchical data of claim 284, wherein said first and second trees having been formed by:

said binary strings being converted to numerals.

286. The set of hierarchical data of claim 284, wherein said binary strings comprise binary edge labeled strings.

287. The set of hierarchical data of claim 276, wherein said strings comprise binary strings, and wherein said second and third string are formed by:

converting said first string to a binary numeral, factoring said binary numeral, and converting the factors to said second and third binary strings.

288. A set of hierarchical data comprising:

a tree having been formed by;

converting a string to a numeral; and

converting said numeral to said tree.

289. The set of hierarchical data of claim 288, wherein said string comprises a binary string.

290. The set of hierarchical data of claim 289, wherein said binary string comprises a binary edge labeled string.

291. The set of hierarchical data of claim 288, wherein said tree comprises a binary edge labeled tree.

292. The set of hierarchical data of claim 288, wherein said tree comprises a binary node labeled tree.

293. The set of hierarchical data of claim 292, wherein said tree having been formed by:

said binary node labeled tree being converted to a numeral node labeled tree.

294. The set of hierarchical data of claim 293, wherein said tree having been formed by:

said numerical node labeled tree being converted to a numeral node labeled tree with at least one null label.

295. The set of hierarchical data of claim 288, wherein said tree having been formed by:

said string being converted to a binary edge labeled string.

296. The set of hierarchical data of claim 288, wherein said tree having been formed by:

said tree being converted to a tree with at least one null label.

297. The set of hierarchical data of claim 288, wherein said tree comprises a binary edge labeled tree, wherein said tree having been formed by:

said binary edge labeled tree being converted to a numeral edge labeled tree.

298. The set of hierarchical data of claim 297, wherein said tree having been formed by:

said binary edge labeled tree being converted to a numeral edge labeled tree with at least one null label.

299. The set of hierarchical data of claim 288, wherein said numeral comprises a binary numeral.

300. The set of hierarchical data of claim 288, wherein said tree having been formed by:

said numeral being converted to a binary numeral.

301. A set of hierarchical data comprising:

a string having been formed by;

converting a tree to a numeral; and

converting said numeral to said string.

302. The set of hierarchical data of claim 301, wherein said string comprises a binary string.

303. The set of hierarchical data method of claim 302, wherein said binary string comprises a binary edge labeled string.

304. The set of hierarchical data of claim 301, wherein said tree comprises a binary edge labeled tree.

305. The set of hierarchical data of claim 301, wherein said tree comprises a binary node labeled tree.

306. The set of hierarchical data of claim 301, wherein said tree comprises a numeral node labeled tree, said string having been formed by:

said numeral node labeled tree being converted to a binary node labeled tree.

307. The set of hierarchical data of claim 301, wherein said tree comprises a numeral node labeled tree with at least one null label, said string having been formed by:

said numeral labeled tree being converted to a numeral node labeled tree without a null label.

308. The set of hierarchical data of claim 301, wherein said tree comprises a numeral edge labeled tree, wherein said string having been formed by:

said numeral edge labeled tree being converted to a binary edge labeled tree.

309. The set of hierarchical data of claim 301, wherein said numeral comprises a binary numeral.

310. An ordered set of data comprising:

a set of strings having been ordered by;

preceding longer binary strings with shorter binary strings; and

ordering binary strings of the same length based at least upon their associated binary values when a binary one is added to the left of the binary string.

311. The ordered set of data of claim 310, wherein said binary strings comprise binary node labeled strings;

and

wherein ordering binary node labeled strings of the same length comprises ordering binary node labeled strings based at least upon the associated binary values.

312. The ordered set of data of claim 311, wherein said binary strings comprise binary edge labeled strings;

and

wherein ordering binary edge labeled strings of the same length comprises ordering binary edge labeled strings based at least upon the associated binary values.

313. A set of hierarchical data comprising:

a tree in an enumeration of trees having been formed by;

positioning said tree at a location k, where k is the product of u and v, such that said tree is formed by a union of trees in positions u and v;

k, u, and v comprising positive natural numerals greater than one.

314. The set of hierarchical data of claim 313, wherein said trees comprise binary edge labeled trees.

315. The set of hierarchical data of claim 313, wherein said trees comprise binary node labeled trees.

316. The set of hierarchical data of claim 313, wherein said trees comprise numeral edge labeled trees,

said tree in said enumeration of trees having been formed by:

converting said numeral edge labeled trees to binary edge labeled trees.

317. The set of hierarchical data of claim 313, wherein said trees comprise numeral node labeled trees,

said tree in said enumeration of trees having been formed by:

converting said numeral node labeled trees to binary node labeled trees.

318. The set of hierarchical data of claim 313, said tree in said enumeration of trees having been formed by:

positioning a tree at location j, where j comprises a non-composite numeral greater than one, such that said tree is formed from a prior tree by attaching a node as a root node.

319. The set of hierarchical data of claim 318, wherein said tree in said enumeration of trees having been formed by:

labeling the edge between said node and said root node.

320. The set of hierarchical data of claim 319, wherein said trees comprise binary edge labeled trees, and said tree in said enumeration of trees having been formed by:

said edge being labeled either 0 or 1.

321. The set of hierarchical data of claim 320, said tree in said enumeration of trees having been formed by:

said edge being labeled with a 0, j being P(2m−

1), where P(2m−

1) is the (2m−

1)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

322. The set of hierarchical data of claim 320, said tree in said enumeration of trees having been formed by:

said edge being labeled with a 1, j being P(2m), where P(2m) is the (2m)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

323. The set of hierarchical data of claim 313, wherein said trees comprise numeral edge labeled trees, and said tree in said enumeration of trees having been formed by:

converting said numeral edge labeled trees to binary edge labeled trees.

324. The set of hierarchical data of claim 313, and further comprising:

a string positioned at location k, said string having been formed based at least in part on a binary value corresponding to k with the left most binary digit omitted.

325. The set of hierarchical data of claim 324, wherein said string comprises a binary string.

326. The set of hierarchical data of claim 325, wherein said binary string comprises a binary edge labeled string.

327. The set of hierarchical data of claim 325, wherein said binary string comprises a binary node labeled string.

328. A hierarchical set of data comprising:

a tree having been formed by at least one of a zero-push and a one-push of another tree.

329. The hierarchical set of data of claim 328, wherein said another tree comprises a binary edge labeled tree.

330. The hierarchical set of data of claim 328, and further comprising:

yet another tree having been formed by at least an N-push of said another tree, where N is a positive natural numeral greater than one.

331. The hierarchical set of data of claim 328, wherein said tree having been formed being converted to a binary edge labeled tree.

332. The hierarchical set of data of claim 331, wherein said tree having been formed being converted to a binary edge labeled tree from a binary node labeled tree.

333. The hierarchical set of data of claim 328, wherein said tree comprises a numeral edge labeled tree.

334. The hierarchical set of data of claim 333, wherein said tree having been formed being converted to a binary edge labeled tree from said numeral edge labeled tree.

335. The hierarchical set of data of claim 333, wherein said numeral edge labeled tree including at least one null label.

336. The hierarchical set of data of claim 328, wherein said tree comprises a numeral node labeled tree.

337. The hierarchical set of data of claim 336, wherein said tree having been formed being converted to a binary edge labeled tree from said numeral node labeled tree.

338. The hierarchical set of data of claim 336, wherein said numerical node labeled tree including at least one null label.

339. The hierarchical set of data of claim 328, wherein said tree being located at position P(2k−

1) in a tree enumeration, said tree having been formed by said zero-push, where P(x) is the Keene enumeration of non-composite numerals on the positive natural numerals, k comprising a natural numeral greater than one.

340. The hierarchical set of data of claim 328, wherein said tree being located at position P(2k) in a tree enumeration, said tree having been formed by said one-push, where P(x) is the Keene enumeration of non-composite numerals on the positive natural numerals, k comprising a natural numeral greater than one.

341. A graphical hierarchy representation comprising:

a linear collection of successively adjacent symbols formed by;

employing a first symbol to represent a node or an edge;

employing additional symbols to represent labels of said node or edge;

wherein the successive order of said symbols provides said graphical hierarchy.

342. The graphical hierarchy representation of claim 341, wherein said first symbol represents a node;

said additional symbols comprising two symbols that respectively represent labels 0 and 1; and

wherein said graphical hierarchy comprises a binary edge labeled tree.

343. The graphical hierarchy representation of claim 342, wherein said linear collection being an edge label symbol provided prior to the symbol for a node associated with the edge that it labels.

344. The graphical hierarchy representation of claim 343, wherein said linear collection being an edge label symbol being provided immediately prior to the symbol for a node associated with the edge that it labels.

345. The graphical hierarchy representation of claim 341, wherein said first symbol represents an edge;

said additional symbols comprising two symbols to respectively represent 0 and 1; and

wherein said graphical hierarchy comprises a binary node labeled tree.

346. The graphical hierarchy representation of claim 345, wherein said linear collection being a node label symbol being provided prior to the symbol for an edge associated with for the node that it labels.

347. The graphical hierarchy representation of claim 346, wherein said linear collection being a node label symbol being provided immediately prior to the symbol for an edge associated with the node that it labels.

348. The graphical hierarchy representation of claim 341, wherein adjacent label symbols represent successively connected nodes or edges.

349. The graphical hierarchy representation of claim 341, wherein non-adjacent label symbols represent nodes or edges that are not successively connected.

350. A hierarchical set of data comprising:

a string in an enumeration of strings having been formed by;

positioning at a location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted, k comprising a positive natural numeral greater than one.

351. The hierarchical set of data of claim 350, wherein said strings comprise binary strings.

352. The hierarchical set of data of claim 351, wherein said binary strings comprise binary edge labeled strings.

353. The hierarchical set of data of claim 351, wherein said binary strings comprise binary node labeled strings.

354. An ordered set of data comprising:

a plurality of trees having been ordered by;

converting said plurality of trees to numerals;

ordering said numerals; and

converting said numerals to said plurality of trees.

355. The ordered set of data of claim 354, wherein said trees comprise binary edge labeled trees.

356. The ordered set of data of claim 354, wherein said trees comprise binary node labeled trees.

357. The ordered set of data of claim 354, wherein said trees comprise numeral edge labeled trees,

said trees being ordered by:

converting said numeral edge labeled trees to binary edge labeled trees.

358. The ordered set of data of claim 354, wherein said trees comprise numeral node labeled trees,

said trees being ordered by:

converting said numeral node labeled trees to binary node labeled trees.

359. The ordered set of data of claim 354, and further comprising a plurality of strings having been ordered by:

preceding binary longer strings with shorter binary strings; and

ordering binary strings of the same length based upon their associated binary values when a binary one is added to the left of the binary string;

wherein a relationship exists between the ordered trees and the ordered binary strings

360. The ordered set of data of claim 359, wherein said binary strings comprise binary node labeled strings;

and

wherein ordering binary node labeled strings of the same length comprises ordering binary node labeled strings based at least in part upon the associated binary values.

361. The ordered set of data of claim 359, wherein said binary strings comprise binary edge labeled strings;

and

wherein ordering binary edge labeled strings of the same length comprises ordering binary edge labeled strings based at least in part upon the associated binary values.

362-1. The apparatus of claim 362, wherein said computing platform comprises means for converting said at least two trees to binary edge labeled trees.

371. An apparatus comprising:

a computing platform;

said computing platform comprising means for converting at least two trees to strings, means for combining said strings, and means for converting the combination of strings to a tree.

372. The apparatus of claim 371, wherein said trees comprise binary edge labeled trees.

373. The apparatus of claim 371, wherein said computing platform comprises means for converting said at least two trees to binary edge labeled trees.

374. The apparatus of claim 371, wherein said at least two trees include at least one null label.

375. The apparatus of claim 374, wherein said computing platform comprises means for converting said at least two trees that include at least one null label to trees without a null label.

376. The apparatus of claim 375, wherein said computing platform comprises means for converting said trees without a null label to binary edge labeled trees.

377. The apparatus of claim 371, wherein said computing platform comprises means for converting said at least two trees to binary node labeled trees.

378. The apparatus of claim 371, wherein said strings comprise binary strings.

379. The apparatus of claim 378, wherein said binary strings comprise binary edge labeled strings.

380. The apparatus of claim 371, wherein said strings comprise binary strings, and wherein said computing platform comprises means for combining said binary strings including means for converting said binary strings to binary numerals, means for multiplying said binary numerals, and means for converting the product to a binary string.

381-2. The apparatus of claim 381, wherein said first tree comprises a binary edge labeled tree.

390. An apparatus comprising:

a computing platform;

said computing platform comprising means for converting a first tree to a first string, means for splitting said first string into at least second and third strings, and means for converting said at least second and third strings to at least second and third trees.

391. The apparatus of claim 390, wherein said first tree comprises a binary edge labeled tree.

392. The apparatus of claim 390, wherein said computing platform comprises means for converting said first tree to a binary edge labeled tree.

393. The apparatus of claim 390, wherein said first tree includes at least one null label.

394. The apparatus of claim 393, wherein said computing platform comprises means for converting said first tree to a tree without a null label.

395. The apparatus of claim 393, wherein said computing platform comprises means for converting said tree without a null label to a binary edge labeled tree.

396. The apparatus of claim 393, wherein said computing platform comprises means for converting said tree without a null label to a binary node labeled tree.

397. The apparatus of claim 390, wherein said computing platform comprises means for converting said first tree to a binary node labeled tree.

398. The apparatus of claim 390, wherein said strings comprise binary strings.

399. The apparatus of claim 398, wherein said computing platform comprises means for converting said binary strings to numerals.

400. The apparatus of claim 398, wherein said binary strings comprise binary edge labeled strings.

401. The apparatus of claim 390, wherein said strings comprise binary strings, and wherein said means for splitting said first string into at least second and third strings comprises:

means for converting said first string to a binary numeral, means for factoring said binary numeral, and means for converting the factors to binary strings.

402. An apparatus comprising:

a computing platform;

said computing platform comprising means for converting a string to a numeral, and means for converting said numeral to a tree.

403. The apparatus of claim 402, wherein said string comprises a binary string.

404. The apparatus of claim 403, wherein said binary string comprises a binary edge labeled string.

405. The apparatus of claim 402, wherein said tree comprises a binary edge labeled tree.

406. The apparatus of claim 402, wherein said tree comprises a binary node labeled tree.

407. The apparatus of claim 406, wherein said computing platform comprises means for converting said binary node labeled tree to a numeral node labeled tree.

408. The apparatus of claim 407, wherein said computing platform comprises means for converting said numerical node labeled tree to a numeral node labeled tree with at least one null label.

409. The apparatus of claim 402, wherein said computing platform comprises means for converting said string to a binary edge labeled string.

410. The apparatus of claim 402, wherein said computing platform comprises means for converting said tree to a tree with at least one null label.

411. The apparatus of claim 402, wherein said tree comprises a binary edge labeled tree, wherein said computing platform comprises means for converting said binary edge labeled tree to a numeral edge labeled tree.

412. The apparatus of claim 402, wherein said computing platform comprises means for converting said binary edge labeled tree to a numeral edge labeled tree with at least one null label.

413. The apparatus of claim 402, wherein said numeral comprises a binary numeral.

414. The apparatus of claim 402, wherein said computing platform comprises means for converting said numeral to a binary numeral.

415. An apparatus comprising:

a computing platform;

said computing platform comprising means for converting a tree to a numeral, and means for converting said numeral to a string.

416. The apparatus of claim 415, wherein said string comprises a binary string.

417. The apparatus of claim 416, wherein said binary string comprises a binary edge labeled string.

418. The apparatus of claim 415, wherein said tree comprises a binary edge labeled tree.

419. The apparatus of claim 415, wherein said tree comprises a binary node labeled tree.

420. The apparatus of claim 415, wherein said tree comprises a numeral node labeled tree, said computing platform comprises means for converting said numeral node labeled tree to a binary node labeled tree.

421. The apparatus of claim 415, wherein said tree comprises a numeral node labeled tree with at least one null label, said computing platform comprises means for converting said numeral labeled tree to a numeral node labeled tree without a null label.

422. The apparatus of claim 415, wherein said tree comprises a numeral edge labeled tree, wherein said computing platform comprises means for converting said numeral edge labeled tree to a binary edge labeled tree.

423. The apparatus of claim 415, wherein said numeral comprises a binary numeral.

424. An apparatus comprising:

a computing platform;

said computing platform comprises means for ordering strings, said means for ordering strings comprising;

means for preceding longer binary strings with shorter binary strings; and

means for ordering binary strings of the same length based at least upon their associated binary values when a binary one is added to the left of the binary string.

425. The apparatus of claim 424, wherein said binary strings comprise binary node labeled strings;

and

wherein said computing platform comprises means for ordering binary node labeled strings of the same length based at least upon the associated binary values

426. The apparatus of claim 424, wherein said binary strings comprise binary edge labeled strings;

and

wherein said computing platform comprises means for ordering binary edge labeled strings of the same length based at least upon the associated binary values.

427. An apparatus comprising:

a computing platform;

said platform comprising means for enumerating trees, said means for enumerating trees comprising means for positioning a tree at location k, where k is the product of u and v, such that said tree is formed by a union of trees in positions u and v;

k, u, and v comprising positive natural numerals greater than one.

428. The apparatus of claim 427, wherein said trees comprise binary edge labeled trees.

429. The apparatus of claim 427, wherein said trees comprise binary node labeled trees.

430. The apparatus of claim 427, wherein said trees comprise numeral edge labeled trees,

wherein said computing platform comprises means for converting said numeral edge labeled trees to binary edge labeled trees.

431. The apparatus of claim 427, wherein said trees comprise numeral node labeled trees,

and wherein said computing platform comprises means for converting said numeral node labeled trees to binary node labeled trees.

432. The apparatus of claim 427, wherein said computing platform comprises means for positioning a tree at location j, where j comprises a non-composite numeral greater than one, such that said tree is formed from a prior tree by attaching a node as a root node.

433. The apparatus of claim 432, wherein said computing platform comprises means for labeling the edge between said node and said root node.

434. The apparatus of claim 433, wherein said trees comprise binary edge labeled trees, said means for labeling said edge comprising means for labeling said edge 0 or 1.

435. The apparatus of claim 434, wherein said edge capable of being labeled zero, j capable of being P(2m−

1), where P(2m−

1) is the (2m−

1)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

436. The apparatus of claim 434, wherein said edge capable of being labeled one, j capable of being P(2m), where P(2m) is the (2m)th non-composite of the Keene enumeration of non-composite numerals on the positive natural numerals.

437. The apparatus of claim 427, wherein said trees comprise numeral edge labeled trees, and wherein said computing platform comprises means for converting said numeral edge labeled trees to binary edge labeled trees.

438. The apparatus of claim 427, wherein said computing platform comprising means for positioning at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted.

439. The apparatus of claim 438, wherein said strings comprise binary strings.

440. The apparatus of claim 439, wherein said binary string comprises a binary edge labeled string.

441. The apparatus of claim 439, wherein said binary string comprises a binary node labeled string.

442. An apparatus comprising:

a computing platform;

said computing platform comprising means for forming at least one of a zero-push and a one-push of a tree.

443. The apparatus of claim 442, wherein said tree comprises a binary edge labeled tree.

444. The apparatus of claim 442, wherein said computing platform comprises means for forming a tree comprising at least an N-push of said tree, where N is a positive natural numeral greater than one.

445. The apparatus of claim 442, wherein said computing platform comprises means for converting said tree to a binary edge labeled tree.

446. The apparatus of claim 445, wherein said computing platform comprises means for converting said tree to said binary edge labeled tree from a binary node labeled tree.

447. The apparatus of claim 442, wherein said tree comprises a numeral edge labeled tree

448. The apparatus of claim 447, wherein said computing platform comprises means for converting said tree to a binary edge labeled tree from said numeral edge labeled tree.

449. The apparatus of claim 448, wherein said numeral edge labeled tree includes at least one null label.

450. The apparatus of claim 442, wherein said tree comprises a numeral node labeled tree.

451. The apparatus of claim 450, wherein said computing platform comprises means for converting said tree to a binary edge labeled tree from said numeral node labeled tree.

452. The apparatus of claim 450, wherein said numerical node labeled tree includes at least one null label.

453. An apparatus comprising:

a computing platform;

said computing platform being adapted to employ a first symbol to represent a node or an edge, and employ additional symbols to represent labels of said node or edge, wherein the successive order of said symbols provides a graphical hierarchy.

454. The apparatus of claim 453, wherein said first symbol represents a node;

said additional symbols comprise two symbols that respectively represent labels 0 and 1; and

wherein said graphical hierarchy comprises a binary edge labeled tree.

455. The apparatus of claim 454, wherein said computing platform comprises means for providing an edge label symbol prior to the symbol for a node associated with the edge that it labels.

456. The apparatus of claim 455, wherein said computing platform comprises means for providing an edge label symbol immediately prior to the symbol for a node associated with the edge that it labels

457. The apparatus of claim 453, wherein said first symbol represents an edge;

said additional symbols comprise two symbols to respectively represent 0 and 1; and

wherein said graphical hierarchy comprises a binary node labeled tree.

458. The apparatus of claim 457, wherein said computing platform comprises means for providing a node label symbol prior to the symbol for an edge associated with the node that it labels.

459. The apparatus of claim 458, wherein said computing platform comprises means for providing a node label symbol immediately prior to the symbol for an edge associated with the node that it labels.

460. The apparatus of claim 453, wherein said computing platform comprise means for providing adjacent label symbols to represent successively connected nodes or edges.

461. The apparatus of claim 453, wherein said computing platform comprises means for providing non-adjacent label symbols to represent nodes or edges that are not successively connected.

462. An apparatus comprising:

a computing platform;

said computing platform comprising means for positioning at location k a string such that said string corresponds to a binary value, said binary value corresponding to k with the left most binary digit omitted.

463. The apparatus of claim 462, wherein said string comprises a binary string.

464. The apparatus of claim 462, wherein said binary string comprises a binary edge labeled string.

465. The apparatus of claim 463, wherein said binary string comprises a binary node labeled string.

466. An apparatus comprising:

a computing platform;

said computing platform comprising means for converting a set of trees to numerals, means for ordering said numerals, and means for converting said numerals to said set of trees.

467. The apparatus of claim 466, wherein said trees comprise binary edge labeled trees.

468. The apparatus of claim 466, wherein said trees comprise binary node labeled trees.

469. The apparatus of claim 466, wherein said trees comprise numeral edge labeled trees,

and wherein said computing platform comprises means for converting said numeral edge labeled trees to binary edge labeled trees.

470. The apparatus of claim 466, wherein said trees comprise numeral node labeled trees,

and wherein said computing platform comprises means for converting said numeral node labeled trees to binary node labeled trees.

471. The apparatus of claim 470, wherein said computing platform comprise means for ordering a set of strings, said means for ordering comprising:

means for preceding longer binary strings with shorter binary strings; and

means for ordering binary strings of the same length based upon their associated binary values when a binary one is added to the left of the binary string;

wherein a relationship exists between the ordered trees and the ordered binary strings.

472. The apparatus of claim 471, wherein said binary strings comprise binary node labeled strings;

and

wherein said computing platform comprises means for ordering binary node labeled strings of the same length based at least in part upon the associated binary values.

473. The apparatus of claim 471, wherein said binary strings comprise binary edge labeled strings;

and

wherein said computing platform comprises means for ordering binary edge labeled strings of the same length based at least in part upon the associated binary values.