Enumeration of trees from finite number of nodes

• US 20060129582A1
• Filed: 12/06/2004
• Published: 06/15/2006
• Est. Priority Date: 12/06/2004
• 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: Andrews, Mark, Schiffmann, Karl, LeTourneau, Jack J.
• Granted Patent: US 7,636,727 B2
• Time in Patent Office: Days
• US Class Current: 1/1
• CPC Class Codes: G06F 16/211 Schema design and management
• CPC Class Codes: G06F 16/2246 Trees, e.g. B+trees
• CPC Class Codes: G06F 16/2264 Multidimensional index structures
• CPC Class Codes: G06F 16/9027 Trees
• CPC Class Codes: Y10S 707/99942 Manipulating data structure, e.g. compression, compaction, compilation

Abstract

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

Claims

1. A method comprising:

enumerating possible trees configurable from a finite number (N) of nodes; and

determining for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals.

2. The method of claim 1, wherein the method further comprises:

determining one or more allocations of the finite number of nodes among subtree slots, an allocation defining a portion of the finite number of nodes to a subtree slot in the allocation;

for a subtree slot in an allocation, enumerating a set containing one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

3. The method of claim 2, wherein the method further comprises pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot.

4. The method of claim 3, wherein the method further comprises merging pushed sets of subtrees in an allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

5. The method of claim 4, wherein the method further comprises determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from the allocations of the finite number of nodes among subtree slots.

6. The method of claim 4, the method further comprising:

representing a subtree in a pushed set of subtrees of the allocation with a natural numeral; and

multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with a natural numeral corresponding to an element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

7. The method of claim 4, the method further comprising multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with an element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

8. The method of claim 1, wherein the enumerated trees comprise binary edge labeled trees.

9. The method of claim 1, the method further comprising:

identifying N-1 arrangements of subtree slots coupled to a root node; and

determining possible allocations of N-1 nodes among subtree slots in arrangements of subtree slots;

for a subtree slot in a possible allocation of the N-1 nodes, determining one or more natural numerals for possible configurations of a subtree from nodes allocated to the subtree slot.

10. The method of claim 9, the method further comprising:

allocating a portion of N-1 nodes to a subtree slot in an arrangement of subtree slots; and

for a subtree slot in the arrangement of subtree slots;

enumerating one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

11. The method of claim 1, wherein said enumerating comprises enumeration of each of one or more possible trees, and wherein said determining a natural numeral for the enumerated trees comprises determining a natural numeral for each one of the enumerated trees.

12. The method of claim 11, the method further comprising:

identifying each of N-1 arrangements of subtree slots coupled to a root node;

determining each possible allocation of N-1 nodes among subtree slots in each arrangement of subtree slots;

for each subtree slot in each possible allocation of the N-1 nodes, determining one or more natural numerals for each possible configuration of a subtree from nodes allocated to the subtree slot.

13. The method of claim 12, the method further comprising:

allocating a portion of N-1 nodes to each subtree slot in each arrangement of subtree slots; and

for each subtree slot in the arrangement of subtree slots;

enumerating each of one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

14. The method of claim 11, wherein the method further comprises:

determining one or more allocations of the finite number of nodes among subtree slots, each allocation defining a portion of the finite number of nodes to a subtree slot in the allocation;

for each subtree slot in each allocation, enumerating a set containing each of one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

15. The method of claim 14, wherein the method further comprises:

pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot; and

merging pushed sets of subtrees in each allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

16. The method of claim 15, wherein the method further comprises multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

17. The method of claim 15, the method further comprising:

representing each subtree in each pushed set of subtrees of the allocation with a natural numeral; and

multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each natural numeral corresponding to each element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

18. The method of claim 15, wherein the method further comprises determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from each of the allocations of the finite number of nodes among subtree slots.

19. The method of claim 1, wherein the natural numeral comprises a product of one or more component natural numerals, the one or more component natural numerals representing a subtree among subtrees which are merged at a root node to form the associated enumerated tree.

20. The method of claim 1, the method further comprising:

identifying N-1 arrangements of subtree slots coupled to a root node;

for an arrangement of subtree slots;

determining one or more allocations of N-1 nodes among the subtree slots in the arrangement, a subtree slot being allocated a portion of said N-1 nodes; and

for the subtree slots, enumerating one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

21. The method of claim 20, the method further comprising:

associating the enumerated one or more possible subtrees for subtree slot with a natural numeral, the natural numeral being based, at least in part, on a said association; and

determining the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

22. The method of claim 21, wherein the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

23. The method of claim 20, wherein the method further comprises:

identifying each enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, each subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association among the allocated nodes, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

24. The method of claim 20, wherein the method further comprises:

identifying an enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, a subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

25. The method of claim 20, wherein:

identifying N-1 arrangements of subtree slots comprises identifying each of N-1 arrangements of subtree slots coupled to the root node;

determining one or more allocations of N-1 nodes comprises determining one or more allocations of N-1 nodes among the subtree slots in the arrangement, each subtree slot being allocated a portion of said N-1 nodes; and

enumerating one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot comprises enumerating each of one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

26. The method of claim 25, the method further comprising:

associating each of the enumerated one or more possible subtrees for each subtree slot with a natural numeral, the natural numeral being based, at least in part, on said association; and

determining the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

27. The method of claim 26, wherein the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

28. The method of claim 1, wherein said enumerating possible configurable trees comprises enumerating the possible trees configurable from exactly N nodes.

29. An apparatus comprising:

means for enumerating possible trees configurable from a finite number (N) of nodes; and

means for determining for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals.

30. The apparatus of claim 29, wherein the apparatus further comprises:

means for determining one or more allocations of the finite number of nodes among subtree slots, an allocation defining a portion of the finite number of nodes to a subtree slot in the allocation;

for a subtree slot in an allocation, means for enumerating a set containing one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

31. The apparatus of claim 30, wherein the apparatus further comprises means for pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot.

32. The apparatus of claim 31, wherein the apparatus further comprises means for merging pushed sets of subtrees in an allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

33. The apparatus of claim 32, wherein the apparatus further comprises means for determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from the allocations of the finite number of nodes among subtree slots.

34. The apparatus of claim 32, the apparatus further comprising:

means for representing a subtree in a pushed set of subtrees of the allocation with a natural numeral; and

means for multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with a natural numeral corresponding to an element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

35. The apparatus of claim 32, the apparatus further comprising means for multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with an element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

36. The apparatus of claim 29, wherein the enumerated trees comprise binary edge labeled trees.

37. The apparatus of claim 29, the apparatus further comprising:

means for identifying N-1 arrangements of subtree slots coupled to a root node;

means for determining possible allocations of N-1 nodes among subtree slots in arrangements of subtree slots; and

for a subtree slot in a possible allocation of the N-1 nodes, means for determining one or more natural numerals for possible configurations of a subtree from nodes allocated to the subtree slot.

38. The apparatus of claim 37, the apparatus further comprising:

means for allocating a portion of N-1 nodes to a subtree slot in an arrangement of subtree slots;

means for enumerating one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

means for performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

39. The apparatus of claim 29, wherein said means for enumerating comprises means for enumerating each of one or more possible trees, and wherein said means for determining a natural numeral for the enumerated trees comprises means for determining a natural numeral for each one of the enumerated trees.

40. The apparatus of claim 39, the apparatus further comprising:

means for identifying each of N-1 arrangements of subtree slots coupled to a root node;

means for determining each possible allocation of N-1 nodes among subtree slots in each arrangement of subtree slots; and

means for determining one or more natural numerals for each possible configuration of a subtree from nodes allocated to the subtree slot.

41. The apparatus of claim 40, the apparatus further comprising:

means for allocating a portion of N-1 nodes to each subtree slot in each arrangement of subtree slots;

means for enumerating each of one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

means for performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

42. The apparatus of claim 39, wherein the apparatus further comprises:

means for determining one or more allocations of the finite number of nodes among subtree slots, each allocation defining a portion of the finite number of nodes to a subtree slot in the allocation;

means for enumerating a set containing each of one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

43. The apparatus of claim 42, wherein the apparatus further comprises:

means for pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot; and

means for merging pushed sets of subtrees in each allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

44. The apparatus of claim 43, wherein the apparatus further comprises means for multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

45. The apparatus of claim 43, the apparatus further comprising:

means for representing each subtree in each pushed set of subtrees of the allocation with a natural numeral; and

means for multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each natural numeral corresponding to each element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

46. The apparatus of claim 43, wherein the apparatus further comprises means for determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from each of the allocations of the finite number of nodes among subtree slots.

47. The apparatus of claim 29, wherein the natural numeral comprises a product of one or more component natural numerals, the one or more component natural numerals representing a subtree among subtrees which are merged at a root node to form the associated enumerated tree.

48. The apparatus of claim 29, the apparatus further comprising:

means for identifying N-1 arrangements of subtree slots coupled to a root node;

means for determining one or more allocations of N-1 nodes among the subtree slots in an identified arrangement of subtree slots, a subtree slot being allocated a portion of said N-1 nodes; and

means for enumerating one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

49. The apparatus of claim 48, the apparatus further comprising:

means for associating the enumerated one or more possible subtrees for subtree slot with a natural numeral, the natural numeral being based, at least in part, on a said association; and

means for determining the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

50. The apparatus of claim 49, wherein the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

51. The apparatus of claim 48, wherein the apparatus further comprises:

means for identifying each enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, each subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association among the allocated nodes, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

52. The apparatus of claim 48, wherein the apparatus further comprises:

means for identifying an enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, a subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

53. The apparatus of claim 48, wherein:

means for identifying N-1 arrangements of subtree slots comprises means for identifying each of N-1 arrangements of subtree slots coupled to the root node;

means for determining one or more allocations of N-1 nodes comprises means for determining one or more allocations of N-1 nodes among the subtree slots in the arrangement, each subtree slot being allocated a portion of said N-1 nodes; and

means for enumerating one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot comprises means for enumerating each of one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

54. The apparatus of claim 53, the apparatus further comprising:

means for associating each of the enumerated one or more possible subtrees for each subtree slot with a natural numeral, the natural numeral being based, at least in part, on said association; and

means for determining the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

55. The apparatus of claim 54, wherein the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

56. The apparatus of claim 29, wherein said means for enumerating possible configurable trees comprises means for enumerating the possible trees configurable from exactly N nodes.

57. An apparatus comprising a computing platform, the computing platform being adapted to:

enumerate possible trees configurable from a finite number (N) of nodes; and

determine for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals.

58. The apparatus of claim 57, wherein the computing platform is further adapted to:

determine one or more allocations of the finite number of nodes among subtree slots, an allocation defining a portion of the finite number of nodes to a subtree slot in the allocation;

for a subtree slot in an allocation, enumerate a set containing one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

59. The apparatus of claim 58, wherein the computing platform is further adapted to push the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot.

60. The apparatus of claim 59, wherein the computing platform is further adapted to merge pushed sets of subtrees in an allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

61. The apparatus of claim 60, wherein the computing platform is further adapted to determine the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from the allocations of the finite number of nodes among subtree slots.

62. The apparatus of claim 60, wherein the computing platform is further adapted to:

represent a subtree in a pushed set of subtrees of the allocation with a natural numeral; and

multiply a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with a natural numeral corresponding to an element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

63. The apparatus of claim 60, wherein the computing platform is further adapted to multiply a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with an element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

64. The apparatus of claim 57, wherein the enumerated trees comprise binary edge labeled trees.

65. The apparatus of claim 57, wherein the computing platform is further adapted to:

identify N-1 arrangements of subtree slots coupled to a root node;

determine possible allocations of N-1 nodes among subtree slots in arrangements of subtree slots; and

for a subtree slot in a possible allocation of the N-1 nodes, determine one or more natural numerals for possible configurations of a subtree from nodes allocated to the subtree slot.

66. The apparatus of claim 65, wherein the computing platform is further adapted to:

allocate a portion of N-1 nodes to a subtree slot in an arrangement of subtree slots;

enumerate one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

perform a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

67. The apparatus of claim 57, wherein the computing platform is further adapted to:

enumerate each of one or more possible trees; and

determine a natural numeral for each one of the enumerated trees.

68. The apparatus of claim 67, wherein the computing platform is further adapted to:

identify each of N-1 arrangements of subtree slots coupled to a root node;

determine each possible allocation of N-1 nodes among subtree slots in each arrangement of subtree slots; and

determine one or more natural numerals for each possible configuration of a subtree from nodes allocated to the subtree slot.

69. The apparatus of claim 68, wherein the computing platform is further adapted to:

allocate a portion of N-1 nodes to each subtree slot in each arrangement of subtree slots;

enumerate each of one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

perform a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

70. The apparatus of claim 67, wherein the computing platform is further adapted to:

determine one or more allocations of the finite number of nodes among subtree slots, each allocation defining a portion of the finite number of nodes to a subtree slot in the allocation; and

enumerate a set containing each of one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

71. The apparatus of claim 70, wherein the computing platform is further adapted to:

push the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot; and

merge pushed sets of subtrees in each allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

72. The apparatus of claim 71, wherein the computing platform is further adapted to multiply each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

73. The apparatus of claim 71, wherein the computing platform is further adapted to:

represent each subtree in each pushed set of subtrees of the allocation with a natural numeral; and

multiply each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each natural numeral corresponding to each element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

74. The apparatus of claim 71, wherein the computing platform is further adapted to determine the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from each of the allocations of the finite number of nodes among subtree slots.

75. The apparatus of claim 57, wherein the natural numeral comprises a product of one or more component natural numerals, the one or more component natural numerals representing a subtree among subtrees which are merged at a root node to form the associated enumerated tree.

76. The apparatus of claim 57, wherein the computing platform is further adapted to:

identify N-1 arrangements of subtree slots coupled to a root node;

determine one or more allocations of N-1 nodes among the subtree slots in an identified arrangement of subtree slots, a subtree slot being allocated a portion of said N-1 nodes; and

enumerate one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

77. The apparatus of claim 76, wherein the computing platform is further adapted to:

associate the enumerated one or more possible subtrees for subtree slot with a natural numeral, the natural numeral being based, at least in part, on a said association; and

determine the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

78. The apparatus of claim 77, wherein the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

79. The apparatus of claim 76, wherein the computing platform is further adapted to:

identify each enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, each subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association among the allocated nodes, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

80. The apparatus of claim 76, wherein the computing platform is further adapted to:

identify an enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, a subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

81. The apparatus of claim 76, wherein the computing platform is further adapted to:

identify each of N-1 arrangements of subtree slots coupled to the root node;

determine one or more allocations of N-1 nodes among the subtree slots in the arrangement, each subtree slot being allocated a portion of said N-1 nodes; and

enumerate one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot comprises enumerating each of one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

82. The apparatus of claim 81, wherein the computing platform is further adapted to:

associate each of the enumerated one or more possible subtrees for each subtree slot with a natural numeral, the natural numeral being based, at least in part, on said association; and

determine the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

83. The apparatus of claim 82, wherein the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

84. The apparatus of claim 57, wherein the computing platform is further adapted to enumerating the possible trees configurable from exactly N nodes.

104. The article of claim 57, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify N-1 arrangements of subtree slots coupled to a root node;

determine one or more allocations of N-1 nodes among the subtree slots in an identified arrangement of subtree slots, a subtree slot being allocated a portion of said N-1 nodes; and

enumerate one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

105. The article of claim 104, wherein the storage medium further comprises machine-readable instructions stored thereon to:

associate the enumerated one or more possible subtrees for subtree slot with a natural numeral, the natural numeral being based, at least in part, on a said association; and

determine the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

106. The article of claim 105, wherein the natural numeral associated with a tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

107. The article of claim 104, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify each enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, each subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association among the allocated nodes, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

108. The article of claim 104, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify an enumerated tree as being a composite of subtrees according to one of the N-1 arrangement of subtree slots, N-1 nodes being allocated among the composite of subtrees, a subtree in the composite of subtrees having a particular configuration of nodes which are allocated to the subtree and being associated with a natural numeral based, at least in part, upon said association, and wherein the natural numeral associated with the tree comprises a combination of the natural numerals associated with the subtrees in the composite of subtrees.

109. The article of claim 104, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify each of N-1 arrangements of subtree slots coupled to the root node;

determine one or more allocations of N-1 nodes among the subtree slots in the arrangement, each subtree slot being allocated a portion of said N-1 nodes; and

enumerate one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot comprises enumerating each of one or more possible subtrees configurable from the portion of said N-1 nodes allocated to the subtree slot.

110. The article of claim 109, wherein the storage medium further comprises machine-readable instructions stored thereon to:

associate each of the enumerated one or more possible subtrees for each subtree slot with a natural numeral, the natural numeral being based, at least in part, on said association; and

determine the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes among subtrees based, at least in part, upon a combination of the natural numerals associated with particular subtrees enumerated from the allocation of N-1 nodes.

111. The article of claim 110, wherein the natural numeral associated with each tree enumerated from an allocation of the N-1 nodes is based, at least in part, upon a product of the natural numerals associated with the particular subtrees enumerated from the allocation of N-1 nodes.

85. An article comprising:

a storage medium comprising machine-readable instructions stored thereon to;

enumerate possible trees configurable from a finite number (N) of nodes; and

determine for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals.

86. The article of claim 85, wherein the storage medium further comprises machine-readable instructions stored thereon to:

determine one or more allocations of the finite number of nodes among subtree slots, an allocation defining a portion of the finite number of nodes to a subtree slot in the allocation; and

for a subtree slot in an allocation, enumerate a set containing one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

87. The article of claim 86, wherein the storage medium further comprises machine-readable instructions stored thereon to push the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot.

88. The article of claim 87, wherein the storage medium further comprises machine-readable instructions stored thereon to merge pushed sets of subtrees in an allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

89. The article of claim 88, wherein the storage medium further comprises machine-readable instructions stored thereon to determine the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from the allocations of the finite number of nodes among subtree slots.

90. The article of claim 88, wherein the storage medium further comprises machine-readable instructions stored thereon to:

represent a subtree in a pushed set of subtrees of the allocation with a natural numeral; and

multiply a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with a natural numeral corresponding to an element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

91. The article of claim 88, wherein the storage medium further comprises machine-readable instructions stored thereon to multiply a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with an element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

92. The article of claim 85, wherein the enumerated trees comprise binary edge labeled trees.

93. The article of claim 85, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify N-1 arrangements of subtree slots coupled to a root node;

determine possible allocations of N-1 nodes among subtree slots in arrangements of subtree slots; and

for a subtree slot in a possible allocation of the N-1 nodes, determine one or more natural numerals for possible configurations of a subtree from nodes allocated to the subtree slot.

94. The article of claim 93, wherein the storage medium further comprises machine-readable instructions stored thereon to:

allocate a portion of N-1 nodes to a subtree slot in an arrangement of subtree slots;

enumerate one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

perform a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

95. The article of claim 85, wherein the storage medium further comprises machine-readable instructions stored thereon to:

enumerate each of one or more possible trees; and

determine a natural numeral for each one of the enumerated trees.

96. The article of claim 95, wherein the storage medium further comprises machine-readable instructions stored thereon to:

identify each of N-1 arrangements of subtree slots coupled to a root node;

determine each possible allocation of N-1 nodes among subtree slots in each arrangement of subtree slots; and

determine one or more natural numerals for each possible configuration of a subtree from nodes allocated to the subtree slot.

97. The article of claim 96, wherein the storage medium further comprises machine-readable instructions stored thereon to:

allocate a portion of N-1 nodes to each subtree slot in each arrangement of subtree slots;

enumerate each of one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot; and

perform a push operation on the enumerated one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N-1 nodes allocated to the subtree slot.

98. The article of claim 95, wherein the storage medium further comprises machine-readable instructions stored thereon to:

determine one or more allocations of the finite number of nodes among subtree slots, each allocation defining a portion of the finite number of nodes to a subtree slot in the allocation; and

enumerate a set containing each of one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.

99. The article of claim 98, wherein the storage medium further comprises machine-readable instructions stored thereon to:

push the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot; and

merge pushed sets of subtrees in each allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.

100. The article of claim 99, wherein the storage medium further comprises machine-readable instructions stored thereon to multiply each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.

101. The article of claim 99, wherein the storage medium further comprises machine-readable instructions stored thereon to:

represent each subtree in each pushed set of subtrees of the allocation with a natural numeral; and

multiply each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each natural numeral corresponding to each element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.

102. The article of claim 99, wherein the storage medium further comprises machine-readable instructions stored thereon to determine the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from each of the allocations of the finite number of nodes among subtree slots.

103. The article of claim 95, wherein the natural numeral comprises a product of one or more component natural numerals, the one or more component natural numerals representing a subtree among subtrees which are merged at a root node to form the associated enumerated tree.

112. The article of claim 85, wherein the storage medium further comprises machine-readable instructions stored thereon to enumerating the possible trees configurable from exactly N nodes.